Printed and bound in Great Britain for Edward Arnold, the educational, academic amI ... London WClB 3DQ by. Biddies Limited, Guildford and King's Lynn.
Second Edition
J.L. Monteith and M.H. Unsworth
,
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Edward Arnold A di\isioll of lloddpr &: SIOlIgilloll
LOND()N NEW YOBI.; MELBOlnjNE AUCKLAND
CONTENTS PREFACE TO THE SECOND EDITION LIST OF SYMBOLS I. SCOPE OF ENVIRONMENTAL PHYSICS
2. GAS LAWS Pressure, volume and temperature Specific heats Lapse rate Water and water vapour Other gases
3. TRANSPORTLAWS General transfer equation Molecular transfer processes Diffusion coefficients Radiation laws
4 RADIATION ENVIRr NMENT ©
Solar radiation Terrestrial radiation Net radiation
1990 J.L. Monteith and M.H. Unsworth
First published in Great Britain 1973 Second edition first published 1990
5. MICROCLIMATOLOGY OF RADIATION Ii) Interception Direct solar radiation Diffuse radiation • Radiation in crop canopies
Distributed in the USA by Routledge, Chapmm) -I F drag force on a particle; ~etention factor . F mass flux of a gas per umt area; flux of radiant energy F g G h
H . I
acceleration by gravity (9.81 m S--2)
flux of heat by conduction, per unit area.
. .
.
.
10- 34 J s); relatIve humIdIty of alf; heIght of cylinder, ClOp, etc. . . total flux of sensible and latent heat, per umt area . intensity of turbulence, i.e. root t;nean. squar~ velOCity/mean velOCity Planck's cnnstant (6.63
X
intensity of radiation (flux per UnIt solId angle)
I
J k k' K
rate of change of stored heat per umt area ..' . von Karman's constant (0.41); thermal cond2~cttvltYI of air; attenuation coefficient. Boltzmann constant (1.38 x 10 JK ) attenuatio~ coefficient; thermal conductivity of a solid. diffusion coefficient for turbulent transfer in air (subscnpts H for heat, M for momentum, V for water vapour, C for CO 2 )
canopy attenuation coefficient
ROMAN ALPHABET A
.'
1"-
ratio of horizontally projected area of an object to its plane or total
surface area
I
mixing length, stopping distance; length of plate in direction of
airstream L L m
M M n N N p p
q
leaf area index' Monin-Obukhov length
flux of long-wa~e radiation per unit area (subscript u for upwards; d for downwards; c from environment; b from body)
mass of a molecule or particle; air mass number rate of heat production by metabolism per unit area of body surface grammolecular mass (subscripts a for dry air: v for water vapour) .
represents a number or dimenSIOnless empmcal constant In several equations Avogadro constant (6.02 x 10"), number of hours of daylight radiance (radiant flux per unit area per umt sold angle)
latent heat equivalent of sweat rate per umt body area total air pressure; interception probahility in hair coat~ specific humidity of air (mass of water vapour per umt mass air)
.' 01 llWlSt
Symbols
x Symbols Q
-.... -...
rate of mass transfer r radius; resistance to transfer (subscripts M momentum, H heat, V water vapour, C for CO2 ); usually applied to boundary layer transfer resistance to transfer m the atmosphere (subscripts M, H, V, C as above) boundary layer resistance of crop for mass transfer canopy resistance thermal resistance of human body thermal resistance of hair, clothing; resistance for forced ventilation in open-top chambers resistance of hole (one side) for mass transfer r, incursion resistance for open-top chambers resistance of pore for mass transfer rp resistance of a set of stomata total resistance of single stoma resistance for heat transfer by convection, i.c. sensible heat resistance for radiative heat transfer (pc';4!TT3) resistance for simultaneous sensible and radiative heat exchange, i.c. rH and rR in parallel ry resistance for water vapour transfer R Gas Constant (8.31 J mol-I K- I) net radiation flux density net radiation absorbed by a surface at the temperature of the ambient aIr s amount of entity per unit mass of air S gas concentration diffuse solar irradiance on horizontal surface solar radiation received by a body, per unit area, as a result of reflection from the environment Sp direct solar irradiance on surface perpendicular to solar beam Sh direct solar irradiance on horizontal surface S, total solar irradiance (usually) on horizontal surface t diffusion pathlength T temperature Ta air temperature To body temperature Tc cloud-base temperature To dew point temperature T, equivalent temperature of air (T + (dy)) T,. apparent equivalent temperature of air (T + (dy')) Tf effective temperature of ambient air Ts. To temperature of surface losing heat to environment Tv virtual temperature T' thermodynamic wet bulb temperature T* standard temperature for vapour pressure specification u optical pathlength of water vapour in the atmosphere u(z) velocity of air at height z above earth's surface u* friction velocity v molecular velocity
x
z
•
XI
deposition velocity sedimentation velocity molar volume at STP (22.4 L) volume vertical velocity of air; depth of precipitable water body weight of animal .. . .. volume fraction (subscripts s for SOIl; I for lIqUid; g for gas), ratIo of cylinder height to radius distance; height above earth's surface roughness length height of equilibrium boundary layer
GREEK ALPHABET a(A) ~
absorption coefficient (subscripts p for photosynthetically active; T for total radiation; r for red; I for mfra-red) absorptivity at wavelength ~ solar elevation; ratio of observed Nusselt number to that for a smooth
plate psychrometer constant (=Cpp/A£) -y apparent v~lue of psychrometer constant ~3 "(rv!r H -y' dry adiabatIC lapse rate, DALR (9,8 X 10 K m ) r depth of a boundary layer . . & rate of change of saturation vapour pressure With temperature, I.e. 6, iJe,( T)iJ T . ratio of molecular weights of water vapour and aIr (0.622) £ apparent emissivity of the atmosphere E, . £(A) emissivity at wavelength ~ angle with respect to solar beam; potential temperature e thermal diffusivity of still air K thermal diffusivity of a solid, e.g. soil wavelength of electromagnetic radiation latent heat of vaporization of water coefficient of dynamic viscosity of air . coefficient of kinematic viscosity of air; frequency of electromagnetic v radiation . reflection coefficient (subscripts p for photosynthetically aClIv~; ~ for P canopy; s for soil; r ~or. red; .i for infra-red; T for total ratliatlon); density of a gas, e.g. air including water vapour component density of dry air p, density of CO, P• density of a liquid PI density of a solid component of soil p, bulk density of soil p' reflection coefficient P ptA) reflectivity of a surface at wavelength A8 2 Stefan-Boltzmann constant (5.67 x 10- W m- K-') the sum of a series flux of momentum per unit area; shearing stress
!
•
I I
, •
,
f
xii
Symbols f~actio~ of incident radiation transmitted, e.g. by a leaf; relaxation time, tIme constant, turbidity coefficient ~ass concentration of CO 2 • c.g. g m- 3 ; angle between a plate and airstream
T
THE SCOPE OF ENVIRONMENTAL PHYSICS
radiant flux density
absolute humidity of air saturated absolute humidity at temperature T CC)
angle of incidence w
angular frequency; solid angle
NON· DIMENSIONAL GROUPS Le Gr Nu Pr Re, Re Ri Sc Sh Stk
Lewis number (KiD) Grashof number Nusselt number Prandtl number (VIK) Roughness Reynolds number (u.zJv) Reynolds number Richardson number Schmidt number (vlD) Sherwood number Stokes number
To grow and reproduce successfully, organisms must come to terms with their environment. Some microorganisms can grow at temperatures between -6 and 100°C and, when they are desiccated, can survive even down to -272°C. Higher forms of life on the other hand have adapted to a relatively narrow range of environments by evolving sensitive physiological responses to external physical stimuli. The physical environment of plants and animals has
five main components which determine the survival of the species: (i)
source of metabolic energy for all forms of life on land and in the
LOGARITHMS In log
the Lovironment is a source of radiant energy which is trapped by the process of photosynthesis in green ccBs and stored in the form of carbohydrates, proteins and fats. These materials arc the primary
(ij)
logarithm to the base e logarithm to the base 10
(iii)
oceans; the environment is a source of the water, nitrogen, minerals and trace elements needed to form the components of living ceBs; factors such as temperature and daylength determine the rates at which plants grow and develop, the demand of animals for food and
the onset of reproductive cycles in both plants and animals; (iv)
the environment provides stimuli, notably in the form of light or gravity, which are perceived by plants and animals and provide
frames of reference both in time and in space. They arc essential for resctting biological clocks, providing a sense of balance, etc.; (v)
the environment determines the distribution and viability of
pathogens and parasites which attack Hving organisms. and the susceptibility of organisms to attack. To understand and explore relationships between organisms and their environment, the biologist should be familiar with the main concepts of the environmental sciences. He must search for links between physiology, •
!
,, •
,,,
, •
,
f, •
biochemistry and molecular biology on the one hand and meteorology, soil
science and oceanography on the other. One of these links is environmental physics-the measurement and analysis of interactions between organisms and their physical environment. The presence of an organism modifies the environment to which it is exposed, so that the physical stimulus received from the environment is partly determined by the physiological response to the environment.
2
The Scope of Environmental Physics
The Scope of Environmental Physics
When an organism interacts with its environment, the physical processes involved are rarely simple and the physiological mechanisms are often imperfectly understood. Fortunately, physicists are trained to use Occam's Razor when they interpret natural phenomena in terms of cause and effect: they observe the behaviour of a system and then seek the simplest way of describing it in terms of governing variables. Boyle's Law and Newton's Laws of Motion are classic examples of this attitude. More complex relations are avoided until the weight of experimental evidence shows they are essential. Many of the equations discussed in this book are approximations to reality which have heen found useful to establish and explore ideas. The art of environmental physics lies in choosing robust approximations which maintain the principles of conservation for mass, momentum and energy" It has become fashionable to describe approximations as models" These models may be either theoretical or experimental and both types are found in this book. To underline the significance of models which are deliberate simplifications of reality, temperature distributions over a real bean leaf, discussed on p. 129, were reproduced on the cover of the book. We have not considered models of plant or animal systems based on computer simulations. They can rarely be tested in the sense that physicists use the word because so many variables and assumptions are deployed in their derivation. Consequently, although they can be useful for identifying the sensitivity of systems to environmental variables, they sddom seem to us to contribute to an understanding of the principles of environmental physics. Several volumes would be needed to review all the principles of environmental physics and the definite article was deliberately omitted from the title of this book because it makes no claim to be comprehensive. However, the topics which it covers are central to the subject: the exchange of radiation, heat, mass and momentum between organisms and their environment. Within these topics, similar analysis can be applied to a number of closely related problems in plant, animal and human ecology. The short bibliography at the end of the book can be consulted for more specialized treatments and for accounts of other branches of the subject such as the physics of soil water. The lack of a common language is often a barrier to progress in interdisciplinary subjects and it is not easy for a physicist or meteorologist with no biological training to communicate with a physiologist or ecologist who is fearful of formulae. Throughout the book, simple electrical analogues are used to describe rates of transfer and exchange between organisms and their environment, and calculus has been kept to a minimum. The concept of resistance has been familiar to plant physiologists for many years, mainly as a way of expressing the physical factors that control rates of transpiration and photosynthesis, and animal physiologists have used the term to describe the insulation provided by clothing, coats, or by a layer of air. in micrometeorology, aerodynamic resistances derived from turbulent transfer coefficients can be used to calculate fluxes from a knowledge of the appropriate gradients, and resistances which govern the loss of water from vegetation are now incorporated in models of the atmosphere that include the behaviour of the earth's surface. Ohm's Law has therefore become an important unifying principle of environmental physics; the basis of a common language for biologists and physicists.
3
The choice of units was dictated by the structure of the Systeme international, modified by retaining the centimetre. For example, the d,mensIOns of leaves are quoted in mm and cm. To adhere stTlctIy to the metre or the millimetre as units of length often needs powers of.I~ to aVOid superfluous zeros and sometimes gives a false impression of precIsion. As most m~asure ments in environmental physics have an accuracy between + 1 and + ~O Yo they should be quoted to 2 or at most 3 significant figures, preferably m a umt chosen to give quantities between 10- 1 and ]0'- The area of a leaf would 3 therefore be quoted as 23.5 cm' rather than 2.35 x 10- m' or 2350 mm'. Conversions from Si to c.g.s. are given in the AppendIx, Table A.!,
1" • •
;
, ;
i,
,, i •
I I
a constant of proportionality, a standard amount of gas is defined by the volume V m occupied by a mole at sta~dard p~essure and temperature (101.3 kPa and 273.2 K) which is 0.0224 m (22.4 htres). Then pVm = RT (2.3)
GAS LAWS
where R = 8.314 ] mol- I K-· I , called the molar gas constant, has the dimensions of a molecular specific heat. . . . Since the pressure exerted by a gas is ~ m~asure of Its kmetlc energy per unit volume, pVm is proportional to the kmettc energy of ~1 ~ole. A mole of any substance contains N molecules where N = 6.02 xIO IS the Avogadro constant. It follows that the mean energy per molecule IS proportiOnal to pVmlN = (RIN)T = kT
where k is the Boltzmann constant. Equation (2.3) is sometimes used in the form
The physical properties of gases are involved in many of the exchanges that take place between organisms and their environment. Therefore the relevant equations for air form an appropriate starting point for an environmental physics text. They also provide a basis for discussing the behaviour of water vapour, a gas whose significance in meteorology, hydrology and ecology is out of all proportion to its relatively small concentration in the atmosphere.
obtained by writing the density of a gas as its molecular mass divided by its molecular volume, i.e. (2.6)
For unit mass of an' gas with volume V, P = IIV so equation (2.5) can also be written in the form
The observable properties of a gas such as its temperature and pressure can be related to the mass and velocity of its constituent molecules by the Kinetic
Theory of Gases which is based on Newton's Laws of Motion. Newton established the principle that when force is applied to a body, its momentum, the product of mass and velocity, changes at a rate proportional to the magnitude of the force. Appropriately, the unit of force in the Systeme International is a newton: and the unit of pressure (force per unit area) is a pascal ~ from the name of another famous natural philosopher. The pressure p which a gas exerts on the surface of a liquid or solid is a measure of the rate at which momentum is transferred to the surface from molecules which strike it and rebound. On the assumption that the kinetic energy of all the molecules in an enclosed space is constant and by making further assumptions about the nature of a perfect gas, a simple relation can be established between pressure and kinetic energy per unit volume. When the 2 density of the gas is p and the mean square molecular velocity is v , the kinetic energy per unit volume is pv 2/2 and (2.1)
implying that pressure is two-thirds of the kinetic energy per unit volume. Although equation (2.1) is central to the Kinetic Theory of Gases, it has little practical value. A number of congruent but more useful relations can be derived from the observations of Boyle and Charles whose gas laws can be combined to give pV", T
(2.5)
p = pRTIM
PRESSURE, VOLUME AND TEMPERATURE
p=pv 2 /3
(2.4)
(2.2)
where V is the volume of a gas at an absolute temperature T (K). To establish
(2.7)
pV= RTIM
Equation (2.7) provides a general basis for exploring the relation between pressure, volume and temperature in unit mass of gas and IS particularly useful in four cases: (1) constant volume - p proportional to T . (2) constant pressure (isobaric) - V proportiOnal to T . (3) constant temperature (isothermal) - V mversely proportiOnal to P (4) constant energy (adiabatic) - p, V and T may all change. When the molecular weight of a gas is known, its density at STP can be calculated from equation (2.6) and its density at any other temperature and pressure from equation (2.5). Table 2.1 contams. the mo~ec~lar weights a~H.t densities at STP of the main constituents of dry au. Mulliplymg each denSity
,I
I,I
Table 2.1
Composition of dry air Per cent
Mass concentration (kg m- 3 )
Gas
Molecular weight (g)
Density at STP (kg m- 3 )
Nitrogen Oxygen Argon Carbon dioxide
28.01 32.00 38.98 44.01
1.250 1.429 1.783 1.977
78.09 20.95 0.93 0.03
0.975 0.300 0.016 0.001
Air
29.00
1.292
100.00
1.292
by volume
,, 6
,
Gas Laws
by the appropriate volume fraction gives the concentration of each component and the sum of these concentrations is the density of dry air. From a J density of 1.292 kg m- and from equation (2.5) the effective molecular weight of air (in g) is 28.94 or 29.00 within 0.2%.
i
Lapse Rate 7
I
Suppose a parcel containing unit mass of air makes a small ascent so that it expands as external pressure falls by dp. If there is no external supply of heat, the energy for expansion must come from cooling of the parcel by an amount dT. It is convenient to treat this as a two-stage process:
SPECIFIC HEATS
(I)
When unit mass of gas is heated but not allowed to expand, the increase in total heat content per unit increase of temperature is known as the specific heat at constant volume, usually given the symbol cv. Conversely, if the gas is allowed to expand in such a way that its pressure stays constant, additional energy is nceded for expansion so that the specific heat at constant pressure c is larger than Cy. p
i
For an adiabatic process, the sum of these quantities must be zero, i.e.
I
cydT + pdV = 0 Differentiating equation (2.7) and putting RIM = c
i
VdP+pdV= (cp-cv)dT
•
i
, I•
(2.10) p
-
Cy
gives
(2. II)
Eliminating pdV from the last two equations gives
I• ,
cpdT= VdP
(2.12)
Integration is now possible by putting V = RTiMp to give dT
R
dp
T
Mc p
p
, •
(2.13)
from which
,I
pdV = (RIM)dT
(2.8) As the difference between the two specific heats is the work done in expansion per unit increase of temperature, it follows that (2.9) The ratio of c p to Cy for gases depends on the energy associated with the vibration and rotation of its molecules and therefore depends on the number of atoms forming the molecule. For diatomic molecules such as nitrogen and oxygen which are the major constituents of air, the theoretical value of crlcv is 7/5, in excellent agreement with experiment. Because c _ Cv = RIM, it p follows that cp = (712)RIM and Cv = (5/2)RIM. In the natural environment, most processes involving the exchange of heat in air OCCur at a pressure (atmospheric) which is effectively constant and since the molecular weight of air is 28.94,
un J g-I
•
•
The work done for a small expansion dV of unit mass of gas at constant pressure p is found by differentiating equation (2.7) to give
Cp = (7/2) x (8.31128.94) =
I I
To evaluate the difference between cp and cv, the work done by expansion can be calculated by considering the special case of a cylinder with crosssection A fitted with a piston which applies a pressure p on gas in the cylinder, thereby exerting a force pA. If the gas is heated and expands to push the cylinder a distance x, the work done is the product of force pA and distance x or pAx which is also the product of the pressure and the change of volume Ax. The same relation is valid for any system in which gas expands at a constant pressure.
(2)
parcel ascends and cools at constant pressure and volume providing energy cydT; parcel expands against external pressure p requiring energy pdV.
(2.14)
where
fi
i i
I ,i
,, i
RI(Mcp ) = (cp
LAPSE RATE
dp = -gpdz
and substituting for p from equation (2.5) gives
Meteorologists apply thermodynamic principles to the atmosphere by imagining that discrete 'parcels' of air are posted either vertically or horizontally by the action of wind and turbulence. Relations between temperature, pressure and height can be deduced by assuming that processes within a parcel are adiabatic, i.e. that the parcel neither gains energy from its environment (e.g. by heating) nor loses energy.
dp
,,
I
cv)lc p = 0.29 for air.
When the change of temperature with pressure for the adiabatic ascent of a parcel has been established, the change of temperature with height can be found when the change of pressure with height is known. A layer of atmosphere with thickness dz and density p exerts a pressure ~p equal to the weight per unit area or gpdz where g is gravitational acceleratIOn. Because p decreases as z increases
I
K-1
-
gM
p
dz
RT
(2.15)
Substitution in equation (2.13) then gives dT --;dz
= -
glc
p
(2.16)
This quantity is known as the Dry Adiabatic Lapse Rate or DALR. When both g and c p are expressed m SI umts, the DALR IS
r
= 9.8 (m s-')/I.OI x lOJ (J kg-I K- 1) ~ I K per lOO m
"Dry' in this context implies that no condensation or evaporation occurs
8
,
Gas Laws
within the parcel. A Wet Adiabatic Lapse rate operates within cloud where the lapse rate is smaller than the DALR because of the release of latent heat by condensation. The difference between the actual lapse rate of air and the DALR is a measure of the vertical stability of the atmosphere. During the day, the lapse rate up to a height of at least I km is usually larger than the DALR and is many times larger immediately over dry sunlit surfaces. Consequently,as": cending parcels rapidly become warmer than their surroundings and experience buoyancy which accelerates their ascent and promotes turbulent mixing so that the atmosphere is said to be unstable. Conversely, temperature usually increases with height at night ('inversion' of the day-time lapse) so that rising parcels become cooler than their environment and further ascent is inhibited by buoyancy. Turbulence is suppressed and the atmosphere is said to be stable.
WATER AND
Water and Water Vapour
,
I
I
much larger volume of saturated water vapour at a smaller pressure es(T). If the work done during this expansion is identified as the latent heat of vaporization, A, and v is the volume of the gas at any point during expansion
Vapour pressure When both air and liquid water are present in a closed container, molecules of water continually escape from the surface into the air to form water vapour but there is a counter-flow of molecules recaptured by the surface. If the air is dry initially, there is a net loss of molecules recognized as 'evaporation' but as the partial pressure (e) of the vapour increases, the evaporation rate decreases, reaching zero when the rate of loss is exactly balanced by the rate of return. The air is then said to be 'saturated' with vapour and the partial pressure is the saturation vapour pressure of water (SVP), often written e,(T) because it depends strongly on temperature. When a surface is maintained at a lower temperature than the air above it, it is possible for molecules to be captured faster than they are lost and this net gain is recognized as 'condensation'. Rigorous expressions for the dependence of es(T) on Tare obtained by integrating the Clausius-Clapeyron equation, but as the procedure is cumbersome, a simpler (and unorthodox) method will be used here with the advantage that it relates vapour pressure to the concepts of latent heat and free energy. Suppose that the evaporation of unit mass of water can be represented by the isothermal expansion of vapour at a fictitious and large pressure eo to a ,
,
,
t',( T)
1\ =
I I
=
e
so that from equation (2.7)
edV = -(RIMw)Tde/e
f
•
(2.17)
edV+ Vde = 0
•
,,
J"
edV
Differentiating equation (2.7) for an isothermal system and putting p • gives
where Mw is the molecular weight of water. Substitution in equation (2.17) then gives
VAPOUR
The evaporation of water at the earth's surface to form water vapour in the atmosphere is a process of major physical and biological importance because the latent heat of vaporization is large in relation to the specific heat of air. The heat released by condensing 1 g of water vapour is enough to raise the temperature of I kg of air by 2.5 K. Water vapour has been called the working substance of the atmospheric heat engine because of its role in global heat transport. On a much smaller scale, it is the amount of latent heat removed by the evaporation of sweat that allows man and many other mammals to survive in the tropics. Sections which follow describe the physical significance of different ways of specifying the amount of vapour in a sample of air and relations between them.
9
1\
= -
RTJdTlde ~ M",~" e
=
RT In[e.!e,(T)]
I •
(2.18)
Mw
Given values of T and es (1) as in Table A.4, the initial pressure eo can now be calculated as
I
(2.19)
I
When a constant value is taken of 1\ = 2048 kJ g-l at lOoC, eo is found to change very little with temperature: it is 2.076 X 10' MPa at O°C and 2.077 X 10' MPa at 20°C. (In fact, the value of 1\ decreases with temperature by about 2.4 J g--l K- 1 and eo decreases with temperature if this is taken into account. Equation (2.19) therefore provides d simple expression for the dependence of e,( T) on T with the form
·
I
e,(T)
=
e" exp (-I\M,JRT)
e,(T') = e" exp (-I\M.)RT')
I !
(2.20)
where eo can be regarded as constant, at least over a restricted temperature range. However, it is more convenient to eliminate eo by normalizing the equation to express es(T) as a proportion of the saturated vapour pressure at some standard temperature T* so that
i
I
(2.21)
•
I
Dividing equation (2.20) by (2.21) then gives
e,(T) = e,( T') exp {A( T- T')IT}
(2.22)
where A = I\MwIRT'. For T' = 273 K (O°C), 1\ = 2470 J g-f K 1 so that A = 19.59. But over the range 273 to 293 K, more exact values of e.eT) (to within I Pal are obtained by giving A an arbitrary value of 19.65 (see Table AA). Similarly, with T' = 293, the calculated value of A is 18.3 but A = 18.00 gives values of e,( T) from 293 to 313 K. (The need to adjust the value of A arises from the slight dependence of eo on temperature coupled with the sensitivity of eJ T) to the value of A).
j ,. , • •
I
10 Cas Laws
e,(T) = eJT') exp {A(T - T')/( T - T')} A
Il
•
An empirical equation introduced by Tetens (1930) has almost the same form as equation (2.22) and is more exact over a much wider temperature range. As given by Murray (1967), it is
where
Water and Water Vapour
I
(2.23)
17.27 T' = 273 K (e,(T') = 0.611 ki'a) T' = 36 K =
Values of saturation vapour pressure from the Tetens formula arc within 1 Pa of the exact values given in Appendix A4 up to 35°C.
The rate of increase of es(T) with temperature is an important quantity in micro meteorology (see Chapter Il) and is usually given the symbol a or s. Between 0 and 30°C, eJT) increases by about 6.5% per °C whereas the pressure of unsaturated vapour of any ideal gas increases by only 0.4% (1/273) per 0C. By differentiating equation (2.20) with respect to T it can be shown that (2.24)
and this expression is exact enough for all practical purposes up to 400C.
I
The dew point (T,d of a sample of air with vapour pressure e is the temperature to which it must be cooled to become saturated, i.c. it is defined by the equation e = e,(T d ). When the vapour pressure is known, the dew point can be found approximately from tables of SVP or more exactly by
I i
8
I
\
,·
50
•
-'" 40 E -.-
,•
, \
i
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T, ,
= --c-~---
-6 0
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a.
0>
x
30'E
>-
~
'0
"
V> V>
4
"
.c
~
0-
20 "
~
"
0
0
a.
U>
~
~
X
~
s _ _ _ _ _ _ _ _ _ _ _ _ _~W!!_---
Td
T
Temperature Fig.2.2 Relation between dew point temperature, saturation vapour pressure deficit and relative humidity of air sample (see text and also Fig. 11.2).
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TRANSPORT LAWS
P{O)
+I
fJP/oz
, I
I
I
I
I I I I
I
/
I
a
I
I
a
,l,
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a
a
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;
I The last chapter was concerned primarily with ways of specifying the state of the atmosphere in terms of properties such as pressure and temperature. To complete this introduction to some of the major concepts and principles which environmental physics depends on, we now consider how the transport of entities such as mass, momentum and energy is determined by the state of the atmosphere and the corresponding state of the surface involved in the exchange, whether soil, vegetation or the coat of an animal. Two main types of transport system are considered: the first depends on the motion of molecules or particles and the second on electromagnetic waves.
TRANSFER EQUATION A simple general equation can be derived for transport within a gas by 'carriers', which may be molecules or particles or eddies, capable of transporting units of a property P such as momentum or heat or mass. Even when the carriers are moving randomly, net transport may occur in any direction provided P decreases with distance in that direction. The carrier can then unload its excess of l' at a point where the local value is less than at the starting point. To evaluate the net flow of P in one dimension, consider a volume of gas with unit horizontal cross-section and a vertical height L assumed to be the mean distance for unloading a property of the carrier (Fig. 3.1). Over the plane defining the base of the volume, l' has a uniform value 1'(0) and if the vertical gradient (change with height) of l' is dpldz, the value at height I will be 1'(0) + ldpldz. Carriers which originate from a height I will therefore have a load corresponding to 1'(0) +ldl'ldz and if they move a distance I vertically downw,lfds to the plane where the standard load corresponds to 1'(0), they will be able to unload an excess of IdPldz. To find the rate of transport equivalent to this excess, assume that II carriers per unit volume move with a mean random velocity c so that the number moving towards one face of a rectangular volume at any instant is nc/6 per unit area. The downward flux of l' (quantity per unit area and per unit time) is therefore given by (nclI6)dPldz. However, there is a corresponding upward flux of carriers reaching the same plane from below after setting off with a load given by l'-ldpldz. Mathematically, the upward flow of a deficit is
I L __ _
, !, •
//
a
o
I
!,
/
/
---
/
. ---------- --------
o
/,/./
(
/
i
a
I
/
o
/
/
/
l,
I
I I I
/
/
o
o
/
/
________ L - -_ _ _ _ _ /
J.
t P{O)
Fi~. 3.1
Volum~ of a.ir with unit cross-section and height I containing n carriers per
umt volume moving With random velocity c (sec text). equivalent to the downward flow of an excess so the net downward flux of P is (3.1 ) F = (ncl/3)dPldz
. For the application of equation (3.1) to the transport of entities by IOdlvldual molecules, the mean velOCity W IS often related to the root mean square mo~ecular velocity c by assuming that, at any instant, one-third of the mol~cules In the system are moving in the z direction so that W = c/3. Mlcrometcorologists, on the other hand, arc concerned with a form of equation (3.1) in which l' is replaced by the amount of an entity per unit mass of aIr, (s) rather than p~~ umt volume (since the latter depends on temperature) and ~he two quantities arc related by nP = ps, where p is air density. The flux equation therefore becomes F = -pw!(dfldz) (3.2) The minus sign i~ needed to indicate that the flux is downwards if s increases upwards; averaglllg bars are a reminder that both wand s fluctuate over a wide range of time-scales as a consequence of turbulence. In this context, the
--
•
2
2
L
18
Molecular Transfer Processes
Transport Laws
quantity I is known as the 'mixing length' for turbulent transport. It is also possible to write the instantaneous values of sand w as the sum of mean values s or IV and corresponding deviations from the mean s' and Wi. The net flux then becomes
pws
= (pw+pw'j(s+s')
- pw's'
Windspeed u in x direction
(3.3)
where 5' and w' are zero by definition and IV is assumed to be zero near the ground when averaging is performed over a period long compared with the lifetime of the largest eddy (say 10 minutes). This relation provides the 'eddy correlation' method of measuring vertical fluxes discussed in Chapter 14.
MOLECULAR
PROCESSES
: E o Q)
..•.. Momentum transfer
I·. .. . . . . . .
g ttl ..., I.
'-:,-::,--
o'"
.-
According to equation (2.1), the mean square velocity of molecular motion in 3 2 5 an ideal gas is v2 = 3p/p. Substitutingp = 10 N m- and p = 1.29 kg m- for Moleair at O°C gives the root mean square velocity as (V2)1I2 = 480 rn cular motion in air is therefore extremely rapid over the whole range of temperatures found in nature and this motion is responsible for a number of processes fundamental to micro meteorology: the transfer of momentum in moving air responsible for the phenomenon of viscosity; the transfer of heat by the process of conduction; and the transfer of mass by the diffusion of water vapour, carbon dioxide and other gases. Because all three forms of transfer are a direct consequence of molecular motion, they are described by similar relationships which will be considered for the simplest possible case of diffusion in one dimension only.
Force on
Force on surface
,-I
Momentum and viscosity When a stream of air flows over a solid surface, its velocity increases with distance from the surface. For a simple discussion of vi~.;osity, the velocity gradient au/az will be assumed linear as shown in Fig. 3.2. A more realistic velocity profile will be considered in Chapter 7. Provided the air is isothermal, the velocity of molecular agitation will be the same at all distances from the surface but the horizontal component of velocity in the x direction increases with vertical distance z. As a direct consequence of molecular agitation, there is a constant interchange of molecules between adjacent horizontal layers with a corresponding vertical exchange of horizontal momentum. The horizontal momentum of a molecule attributable to motion of the gas as distinct from random motion is mu so from equation (3.1) the rate of transfer of momentum, otherwise known as the shearing stress, can be written (3.4) T = (ncl/3)d(mu)/dz = (cl/3)d(pu)/dz as the density of the gas is p = mn. This is formally identical to the empirical equation defining the kinematic viscosity of a gas v, viz. = vd(pll )fdz
I
the change of p with distance is small, it is more convenient to write .
T
= fLdu/dz
(3.6)
where f.L = pv IS the coefficient of dY na . . . . - . By convention the flux of m()me t ~lllC viSCOSity and P IS a mean density. , n urn IS taken as ·f h .. . 1 toward') a surface and it therefor h th . POS Ive w en It IS dIrected (see Fig. 3.2). e as e sa:ne SIgn as the velocity gradient The momentum transferred la er b I absorbed by the surface which the!ef y ayer through the gas is finally III the direction of the flow Th o.re expefl.ences a ffictlOnal force acting Third Law is the frictional d·rag :x~~~~~~n :~ thIS f~rceh required by Newton's opposite to the flow. n e gas y t e surface m a direction
,, :,:
,, ~
,
I, "
" •
,
Heat and thermal conductivity
,,•
~h~'fgon~~Ctla·onl.jOyferheoaft in still air is aknalogous to the transfer of momentum
. ., ( warm aIr rna es t t .h . velocity of molecules therefore increases ~~ha~. ;It af cooler surface. The IS ance . r~m the surface and the exchange of molecules betwee d' net t~ansfer of molecular energy ann~ ~~~~~t l~yers of air IS responsible for a heat IS proportional to the gradient f h 0 heat. The. rate of transfer of and may therefore be written 0 eat content per Untt volume of the air C = -Kd(pc"T)/dz
(3.5)
showing that v is a function of molecular velocity and mean free path. Where
T(O)
Fig. 3.2 Transfer of momentum from '. .. . related forces. movmg air to a statIOnary surface, showing
_
T
19
where
K,
,·,
(3.7)
the thermal difTusivity of air, has the same dimensions (L2 T- 1) as ,
, --
..
III III
.. .. .. .. .. .. ..
• • • •.. .. ..
..
U1JJUSIOn Coejjicients
20
Transport Laws
I, Warm air
U'l
.: .. ,
)
',-.'
....
E o
":,,:: .::
-'" -''""
•..• Vapour E = -D
~
E o ~
"-'.
'"c
air
: :':
u
~
Humid
~ Concentration 7X~ ccc-
•
Temperature T
''"" -'"
Ll
-"".
:- '-"
,-
-
u
Heat C = -k transfer
::::: transfer
u
aT
co
az
I
u;'-"
o
ax
az
o
: :-:
t Cool surface
, Fig.3.4 Transfer of vapour from humid air to an absorbing surface .
Fig.3.3 Transfer of heat from stilL warm air to a cool surface .
the kinematic viscosity and pCp is the heat content per unit volume of air relative to the value at T = 0 (Oe or K). As in the treatment of momentum, it is convenient to assume that p has a constant value of p over the distance considered and to define a thermal conductivity as k = PCpK so that (3.8) C = -kdTldz identical to the equation for the conduction of heat !n solids. .. In contrast to the convention for momentum, C IS taken as posltlve when the flux of heat is away from the surface in which case dT/dz is negative. The equation therefore contains a minus sign.
Mass transfer and In the presence of a gradient of gas concentration, molecular agitation is responsible for a transfer of mass, generally referred to as 'diffusion' although this word can also be applied to momentum and heat. In Fig. 3.4, a layer of still air containing water vapour makes contact with a hygroscopic surface where water is absorbed. The number of molecules of vapour per unit volume increases with distance from the surface and the exchange of molecules between adjacent layers produces a net movement towards the surface. The transfer of molecules expressed as a mass flux per unit area E is proportional to the gradient of concentration and the transport equation analogous to equations (3.6) and (3.8) is E = -DdX/dz = -Dd(pq)/dz = -pDdq/dz (3.9)
I
where p is an appropriate mean density, D (dimensions L' T- ) is the molecular diffusion coefficient for water vapour and other symbols are defined on p. 11. The sign convention in this equation is the same as for heat.
DIFFUSION COEFFICIENTS Because the same process of molecular agitatioa is responsible for all three types of transfer, the diffusion coefficients for momentum, heat, water vapour and other gases are similar in size and in their dependence on temperature. Values of the coefficients at different temperatures calculated from the Chapman-Eskog kinetic theory of gases agree well with measurements and are given in Table A.3 (p. 267). The temperature dependence of the diffusion coefficients is usually expressed by a power law, e.g. D(T) = D(O){7'lT(O)}" where D(O) IS the coeffiCIent at a base temperature T(O) (K) and n is an index between 1.5 and 2.0. Within the limited range of temperature relevant to environmental physics, say -10 to 50°C, a simple temperature coefficient of 0.007 is accurate enough for practical purposes, i.e. D(T)/D(O)
= K(T)/K(O) = v(T)/v(O) = (1 +O.007T)
where Tis the temperature in °C and the coefficients in units of m v(O) = 13.3 X K(O) = 18.9 x D(O) = 21.2 X = 12.9 x = 11.2 x
10-6 10- 6 10- 6 10- 6 10- 6
(momentum) (heat) (water vapour) (carbon dioxide) (sulphur dioxide)
2
S-l
are
..
22
Transport Laws
Diffusion Coefficients
Resistances Equations (3.4), (3.6) and (3.7) have the same form flux = diffusion coefficient x gradient which is a general way of stating Fick's Law of Diffusion. This law can be applied to problems in which diffusion is a onc-, two- or three-dimensional process but only one-dimensional cases will be considered in this book. Because the gradient of a quantity at a point is often difficult to estimate accurately, Fick's law is generally applied in an integrated form. The integration is very straightforward in cases where the (one-dimensional) flux can be treated as constant in the direction specified by the coordinate z, c.g. at right angles to a surface. Then the integration of (3.9) for example gives E = pq(z,) - PQ(Z2)
.. ..
• •..
..
=
'v 'c
for for for for
,•,
, ,
I
r
Brownian motian The zig-zag motion of particles suspended in a fluid or gas was first described by the English botanist Brown in 1827, but it was nearly 80 years later that Einstein used the kinetic theory of gases to show that the motion was the result of multiple collisions with the surrounding molecules. He found that the mean square displacement X2 of a particle in time t is given by
potential difference across resistance
x 2 = 2Dt
.
resistance
potential difference -7- resistance
Diffusion coefficients have dimensions of (Iength)2 x (time) so the corresponding resistances have dimensions of (time)/(length) or lI(velocity). In a system where rates of diffusion arc governed purely by molecular processes, the coefficients can usually be assumed independent of z so th3t i~:' dzlD, for example, becomes simply (Z2 - z,)1D or (diffusion pathlength) -;- (diffusion ~oefficientr When the molecular diffusion coefficient f?r water v~po~r in air IS 0.25 em S-I, the resistance for a pathlength of 1 em IS 4 s em . It IS often convenient to treat the process of diffusion in laminar boundary layers in terms of resistances and in Chapters 6, 8 and 9, the following symbols are used: resistance resistance resistance resistance
•
,
momentum T = pu -7- Jdz/v heat C = pCp T -;- Jdz/K mass E = pq -;- fdzlD
'M 'II
i
• •
Equivalent expressions for diffusion can be written as follows: rate of transfer of entity
i•
The concept of resistance is not limited to molecular diffusion but is applicable to any system in which fluxes are uniquely related to gradients. In the atmosphere where turbulence is the dominant mechanism of diffusion, diffusion coefficients are several orders of magnitude larger than the corres· ponding molecular value and increase with height above the ground (Chapter 7). Diffusion resistances for momentum, heat, water vapour and carbon dioxide in the atmosphere will be distinguished by the symbols raM, 'aH, 'aV and 'ae; the measurement of these resistances is discussed in Chapters 7 to 9. In studies of the deposition of radioactive material from the atmosphere to the surface, the rate of transfer is sometimes expressed as a deposition velocity which is the reciprocal of a diffusion resistance. In this case the surface concentration is often assumed to be zero and the deposition velocity is found by dividing the dosage of radioactive material by its concentration at an arbitrary height.
,
where pq(zd and PQ(Z2) are concentrations of water vapour at distances z, and Z2 from a surface absorbing or releasing water vapour at a rate E. Usually pq(z,) is taken as the concentration at the surface so that z, = O. Equation (3.10) and similar equations derived by integrating equations (3.6) and (3.8) are analogous to Ohm's Law: current through resistance =
f
(3.10)
" dz/D J"
..
..
i
23
momentum transfer at the surface of a body convective heat transfer water vapour transfer CO 2 transfer
(3.11)
where D is a diffusion coefficient (dimensions L2 T-') for the particle, analogous to the coefficient for gas molecules. The quantity D depends on the intensity of molecular bombardment (a function of absolute temperature), and on the viscosity of the fluid, as follows . Suppose that particles, each with mass m (kg), are dispersed in a container where they neither stick to the walls nor coagulate. The Boltzmann statistical description derived from kinetic theory requires that, as a consequence of the earth's gravitational field, the particle concentration n should decrease exponentially with height z(m) according to the relation
n where
=
nCO) exp (-mgzlkT)
=
dz
= n--:-C= kT
I,.'~ ,
!
: I
If a horizontal area at height z within the container is considered, the flux of particles upwards by diffusion (cf. diffusion of gases) is: F,
I!
,
absolute temperature (K) k = Boltzmann's constant (J K-')
Dmg
J
!,
=
dn -D-
,1
(3.12)
n = nCO) at z = () g = gravitational acceleration (m s· 2) T
\
,, 1
(3.13)
,
•
from equation (3.12). But because all particles tend to move downwards in
!
24
Radiation Luws
Tranjport Laws
response to gravity, there must be a downward flux through the area of F2 = nV~ where V, is the 'sedimentation velocity' (see Chapter 10). Since the system is in equilibrium, F j = F2 and so
D = kTVJmg
(3.14)
For spherical particles, radius r, obeying Stokes' Law, the downward force mg due to gravity is balanced by a drag force 6TIfLrV" and so
D = k716TIfLr (3.15) Thus D depends inversely on particle radius (Table A.6); its dependence on temperature is dominated by the rapid decrease in dynamic viscosity with increasing temperature. Equations (3.11) and (3.15) show that the root mean square displacement iI 5 of a particle by Brownian motion is proportional to T . and to r-0.5. Surprisingly, x 2 does not depend on the mass of the particle, a result confirmed by experiment.
RADIATION LAWS The origin and nature af radiation Electromagnetic radiation is a form of energy derived from oscillating magnetic and electrostatic fields and is capable of transmission through empty space where its velocity is c = 3 X 108 m S-I. The frequency of oscillation v is related to the wavelength A by the standard wave equation c = AV and the wave number I/A = vic is sometimes used as an index of frequency. The ability to emit and absorb radiation is an intrinsic property of solids, liquids and gases and is always associated with cle anges in the energy state of atoms and molecules. Changes in the energy state of atomic electrons are associated with line spectra confined to a specific frequency or set of frequencies. In molecules. the energy of radiation is derived from the vibration and rotation of individual atoms within the molecular structure and numerous possible energy states allow radiation to be emitted or absorbed over a wide range of frequencies to form band spectra. The principle of energy conservation is fundamental to the material origin of radiation. The amount of radiant energy emitted by an individual atom or molecule is equal to the decrease in the potential energy of its constituents.
Full or black body radiation Relationships between radiation absorbed and emitted by matter were examined by Kirchhoff. He defined the absorptivity of a surface "(A) as the fraction of incident radiation absorbed at a specific wavelength A. The emissivity E(A) was defined as the ratio of the actual radiation emitted at the wavelength A to a hypothetical amount of radiant flux B(A). By considering the thermal equilibrium of an object inside an enclosure at a uniform temperature, he showed that "(A) is always equal to E(A). For an object
I •
I I I,
, .
i
25
completely absorbing radiation at wavelength A, "(A) = I, E(A) = I and the emitted radiation is B(A). In the special case of an object with £ = 1 at all wavelengths, the spectrum of emitted radiation is known as the 'full' or black body spectrum. Within the range of temperatures prevailing at the earth's surface, nearly all the radiation emitted by full radiators is confined to the waveband 3 to 100 !-lm, and most natural objects soil, vegetation, waterbehave radiatively almost like full radiators in this restricted region of the spectrum (but not in the visible spectrum). Even fresh snow, one of the whitest surfaces in nature, emits radiation like a black body between 3 and 100 j-lm. The statement 'snow behaves like a black body' refers therefore to the radiation emitted by a snow surface and not to solar radiation reflected by snow. The semantic confusion inherent in the term 'black body' can be avoided hy referring to 'full radiation' and to a 'full radiator'. After Kirchhoff's work was published in 1859, the emission of radiation hy matter was investigated by a number of experimental and theoretical physi· cists. By combining a spectrometer with a sensitive thermopile, they established that the spectral distribution of radiation from a full radiator resembles 0 the curve in Fig. 3.5 in which the chosen temperatures of 6000 and 300 K correspond approximately to the black body temperatures of the sun and the earth's surface.
Wien's Law Wi en deduced from thermodynamic principles that the energy per unit wavelength E(A) should be a function of absolute temperature T and of A such that (3.16) E(A) = J(AT)/~' Deductions from this relation are: (I)
(2)
that when spectra from full radiators at different temperatures are compared, the wavelength Am at which E(A) reaches a maximum should be inversely proportional to T so that ~mT is the same for all values of T. The value of this constant is 2897 fLm K so, in Fig. 3.5, Am = 0.48 fLm for a solar spectrum (T= 6000 K) and 9.7 fLm for a terrestrial spectrum (T= 300 K); the value of E(A) at 'm is proportional to Am -, and therefore to T' because Am?: lIT. The ratio of E(Am} for solar and terrestrial radiation is therefore (6000/300)' = 3.2 x 10".
Stefan's Law Stefan and Boltzmann showed that the energy emitted by a full radiator was proportional to the fourth power of its absolute temperature: in symbols, B=~T'
(3.17)
where n (W 111-. 2 ) is the flux emitted by unit .Hca of a plane surface into an imaginary hemisphere surrounding it and the Stefan-Boltzmann constant is
26
Radiation Laws
Transport Laws
much of modern physics by introducing the quantum hypothesis. Planck found that the spectrum could not be predicted from classical mechanics because its principles imposed no restriction on the amount of radiant energy which a molecule could emit. He postulated that energy was emitted in discrete packets which he called 'quanta' and that the energy of a single quantum Eq was proportional to the frequency of the radiation, i.e.
A (I'm) (T=300Kl
4
-E
10c
12
8,
16
,
20
24
'
28
,
/.h,... .
::l
30
I
E ::l
I I
Eq
I !
I I
I
- 20 -
!
,
c '" '"> '"
. I I I
'"~
I I
!
I
~
'"c '" W ~
O~~··-·--~--+~'~~,~------~--~~----~,~ o 0.4
(3.IX)
I
= ---;-:-.".-;-=------:c
exp {hcl(kA T) - I}
(3.19)
where k is the Boltzmann constant (see p. 5). Differentiation of this equatIOn with respect to A gives the constant of Wien's Law and integration reveals that the Stefan· Boltzmann constant is a multiple of k"c·-1h- J •
•
I
02
,•
I
10
I
~
E ( A)
~
Am
I I
hv
87ThCA -5
.c
I .
c => c
=
where h = 6.63 X 10--.'14 J s is Planck's constant. The formula derived by Planck on the basis of this hypothesis was
I
-
27
0.6
0.8
1.0
1.2
I
Quantum unit
,
Because the amount of energy in a single quantum is inconveniently small, photochemists express the number of quanta in a reaction as a multiple of the Avogadro constant N = 6.02 X 1023. This number of quanta was originally called an 'Einstein' in recognition of Einstein's contribution to the founda· tions of photochemical theory. It is now usually called a mole. (Although the mole is formally defined as 'an amount of substance', its application in this context suggests that it should now be redefined simply as 'a number, namely, the number of molecules in 0.012 kg of carbon 12'.) A photochemical reaction requiring one quantum per molecule of a compound therefore requires one mole of quanta per mole of ~ompound.
I,
14
A (fLm) (T=6000Kl Fig. 3.5 Spectral distribution of radiant energr from a f~J1 radiator at a te":lperature of (a) 6000 K, left-hand vertical and lower hOflzontai aXIs a?d (b) 300 K, nght-hand vertical and upper horizontal axis. About lO?o of, the en~rgy IS emitted at wave.lengths longer than those shown in the diagram. If thiS tali wer~ mcluded, the total area. under the curve would be proportional to crT4 (W m- 2 ). ~m IS the ,v3velength at which the energy per unit wavelength is maximal.
where n is a numerical index. For a 'grey' surface whose emissivity is .almost independent of wavelength, n = 4. When radiation is emitted predominantly at wavelengths less than ~m' n exceeds 4 and conversely when the bulk of emitted radiation appears in a waveband above Am' n IS less than 4.
Planck's Law Prolonged attempts to establish the shape of the black-body spectrum culminated in a theory developed by Max Planck who latd the foundalton for
, .i I
ji
II i,"'
Small temperature
5.67 X 10 -8 W m -2 K -I. The spatial distribution of this nux is considered on pp. 2&--30. . ... d f . ·t As a generalization from equatIOn \3. ~ ~), the radiatIOn cmltte. ro~ unt area of any plane surface with an emissIvity of E«1) can be wntten m the form 1> = EIT Til
,,
,
Although the total radiation emitted by a full radiator is proportional to T4, the exchange of energy between radiators at temperatures TI and T2 becomes nearly proportional to (T2 - Td when this difference is small, a common case in natural environments. If we write (T, - Til = oT, the difference between full radiation at the two temperatures is
R
=
,,{(T , +oT)'- T ,'}
=
. The nature of reflection from the surface of an object depends in a complex way on its electrical properties and on the structure of the surface. In general, specular reflection asSUmes increasing importance as the angle of incidence increases and surfaces acting as specular reflectors absorb less radiation than diffuse reflectors made of the same material. Most natural surfaces act as diffuse reflectors when \fI is less than 60 or 700, but as tV approaches 90°, a condition known as grazing incidence, the reflection from open water, waxy leaves and other smooth surfaces becomes dominantly specular and there is a corresponding increase in reflectivity. The effect is often visible at sunrise and sunset over an extensive water surface, or a lawn, or a field of barley in car.
Radiance and irradiance independent of 4>. On the other hand, the flux per unit solid angle divided by the true area of the surface must be proportional to cos 4>. Figure 3.7 makes this point diagrammatically. A radiometer R mounted vertically above an extended horizontal surface XY 'sees' an area dA and measures a flux which is proportional to dA. When the surface is tilted through an angle ~, the radiometer now sees a larger surface dAlcos but provided the temperature of the surface stays the same, its radiance will be constant and the flux recorded by the radiometer will also be constant. It follows that the flux emitted per unit area (the emittance of the surface) at an angle 4> must be proportional to cos 4> so that the product of emittance (oc cos 4» and the area emitting to the instrument (oc Ilcos 4» stays the same for all values of 4>. This argument is the basis of Lambert's Cosine Law which states that when radiation is emitted by a full radiator at an angle 4> to the normal, the flux per unit solid angle emitted by unit surface is proportional to cos 4>. As a corollary to Lambert's Law, it can be shown by simple geometry that when a full radiator is exposed to a beam of radiant energy at an angle \fI to the normal, the flux density of the absorbed radiation is proportional to cos 1jJ. In remote
*,
When a plane surface is surrounded by a uniform source of radiant energy, a simple relation exists between the irradiance of the surface (the flux incident per unit area) and the radiance of the source. Figure 3.8 displays a surface of unit area surrounded by a radiating hemispherical shell so large that the surface can be treated as a point at the centre of the hemisphere. The area dS (black) is a small element of radiating surface and the radiation reaching the centre of the hemisphere from dS makes an angle fl with the normal to the plane. As the projection of unit area in the direction of the radiation is I X cos fl, the solid angle which the area subtends at dS is w = cos fl/? If the element dS has a radiance N, the flux emitted by dS in the direction of the disc must be N x dS x w = NdS cos fl/? To find the total irradiance of the disc, this quantity must be integrated over the whole hemisphere, but if the radiance is uniform, conventional calculus can be avoided by noting that dS cos fl is the area dS projected on the equatorial plane. It follows that J cos j3 dS is the area of the whole plane or 'IT? so that the total irradianee at the centre of the plane becomes (Nlr) I cos f>dS = 'ITN
(3.23)
The irradiance expressed in watts m- 2 is therefore found by multiplying the
32
Radiation Laws
Transport Laws
through a gas or liquid, quanta encounter molecules of the medium or particles in suspension. After interception, a quantum may suffer one of two fates: it may be absorbed, thereby increasing the energy of the absorbing molecule or particle; or it may be scattered, i.c. diverted from its previous course either forwards (within 900 of the beam) or backwards in a process akin to reflection from a solid. After transmission through the medium, the beam is said to be 'attenuated' by losses caused by absorption and scattering. Beer's Law, frequently invoked in environmental physics, describes attenuation in a very simple system where radiation of a single wavelength is absorbed but not scattered when it passes through a homogeneous medium. Suppose that at some distance x into the medium the flux density of radiation is (x) (Fig. 3.~). Ahsorption in a thin layer dx, assumed proportional to '"o ~
.-c
" '"c. o'" c ~
Distribution of energy in the spectrum of radiation emitted by the sun Energy Waveband ('Yo) (nm) 1.2 0- 300 7.X 300- 400 (ultra-violet) 39.8 4()(}... 700 (visible/PAR) 38.8 700-1500 (near infra-red)
Table 4.1
12.4
1500-00
o "0 o ~ ~
'"
--
•
.->
I !
o
I
100.0 1.5
0.5
i
! ,,i
Wavelength (fLm) The ultra-violet spectrum can be further subdivided into UVA (400-320 nm) which produces tanning in skin; UVB (320 to 290 nm) responsible for skin cancer and Vitamin 0 synthesis; and UVC (290 to 200 nm), potentially harmful but absorbed almost completely by ozone in the stratosphere. The waveband to which the human eye is sensitive ranges from blue (400 nm) through green (550 nm) to red (700 nm). Photosynthesis is stimulated by radiation in the same waveband which is therefore referred to as Photosynthetically Active Radiation (PAR or PhAR). a misnomer because it is green cells which are active, not radiation. Initially, PAR was applied to 2 radiation measured in units of energy flux density (W m- ) but it is 2 increasingly used for the corresponding quantum flux c."nsity (mol m- S-I) because, when photosynthesis rates are compared for light of different quality (e.g. from the sun and from lamps), they are more closely related to the quantum content than to the energy content of the radiation. The fraction of PAR to total energy in the extraterrestrial solar spectrum is 0.45 (Moon,
1940). Many developmental processes in green plants have been found to depend on the state of the pigment phytochrome which absorbs radiation in wavebands centred at 660 and 730 nm. Physiologists and ecologists now regard the ratio of spectral irradiance at these wavelengths, known as the red: far-red ratio, as an environmental signal of major significance for germination, survival and reproduction (Smith and Morgan, 1981).
IN THE As the solar beam passes through the earth's atmosphere, it is modified in quantity, quality and direction by processes of scattering and absorption (Fig. 4.1). Scattering has two main forms. First, individual quanta striking molecules of any gas in the atmosphere are diverted more or less uniformly in all
,I' i,.,
Fig. 4.1 Successive processes attenuating the solar beam as it penetrates the atmosphere. A-extraterrestrial radiation, B-·after ozone absorption, C-after molecular scattering, D-after aerosol scattering, E-after water vapour and oxygen absorption (from Henderson, 1977).
•
! i
directions, a process known as Rayleigh scattering after the physicist who showed theoretically that the effectiveness of molecular scattering was proportional to the inverse fourth power of the wavelength. The scattering of hlue light (A ~ 400 nm) therefore exceeJs the scattering of red (700 nm) by a factor of (7/4)" or ahout 9. This is the physical basis of the blue colour of the sky as seen from the ground and the blue haze surrounding the Earth photographed by astronauts travelling to the Moon. The apparent redness of the sun's disc is further evidence that blue light has been removed from the beam preferentially. Rayleigh scattering is confined to systems in which the diameter of the scatterer (d) is much smaller than the wavelength of the radiation. This condition is not met for particles of dust, smoke, pollen etc. in the atmosphere, referred to as 'aerosol', for which d and A often have the same order of magnitude. Theory developed hy Mie predicts that the wavelength dependence of scattering should be a function of dlA in this case and that, for some values of the ratio, longer wavelengths should be scattered more than short-the reverse of Rayleigh scattering. This happens rarely, but in 1951, smoke with the appropriate size of particles drifted across the Atlantic from forest fires in Canada and caused much anxiety in Europe when sun and moon turned pale blue! Usually, aerosol contains such a wide range of particle sizes that scattering is not strongly dependent on A and a set of measurements in the English 2 Midlands gave a wavelength dependence between XI . 3 and A (McCartney and
,
•
.., ·: ,I',. I, •
,•
I •
40
Solar Radiation at the Ground
Radiation Environment
Unsworth, 1978). In these measurements, the scattering was predominantly 'forwards', i.e. in the direction of the incident radiation, a characteristic of all scattering by aerosol. The second process of attenuation is absorption by ozone (particularly in the ultra-violet spectrum), by water vapour (with strong bands in the infra-red), and by carbon dioxide, and oxygen. In contrast to scattering, which simply changes the direction of radiation, absorption removes energy from the beam so that the atmosphere is heated, In the visible region of the spectrum, absorption by atmospheric gases is much less important than scattering in determining the spectral distribution of solar energy. In the infra-red spectrum, however, absorption is much more important than scattering and several atmospheric constituents absorb strongly, notably water vapour with absorption bands between 0.9 and 3 fLm. The presence of water vapour in the atmosphere increases the amount of visible radiation relative to infra-red radiation. The extent of absorption and scattering in the atmosphere depends partly on the pathlength of the solar beam and partly on the amount of the attenuating constituent present in the path. The pathlength is usually specified in terms of an air mass number which is the length of the path relative to the depth of the atmosphere. For values of zenith angle IjJ less than 80°, the air mass number is simply m = sec tV, but for values between 80° and 90°, m is smaller than sec tV, because of the earth's curvature and values, corrected for refraction, can be obtained from tables (e.g. List, 1966). The amount of water vapour in the atmosphere can be specified by a depth of 'precipitable water' defined as the amount of rain that would be formed if all the water vapour were condensed (between 5 and 50 mm at most stations). If the precipitable water is u, the path length for water vapour is urn. The amount of ozone is specified by an equivalent depth of the pure gas at a pressure of one atmosphere (101.3 kPa). Clouds, consisting of water droplets or ice crystal~ ,scatter radiation both forwards and backwards but, when the depth at cloud is substantial, back-scattering predominates and thick stratus can reflect up to 70% of incident radiation. appearing snow-white from an aircraft flying above it. About 20% of the radiation may be absorbed leaving only 10% for transmission so that the base of such a cloud seems grey. At the edge of a cumulus, however, where the concentration of droplets is small, forward scattering is strong-the silver lining effect-and under a thin sheet of cirrus, the reduction of irradiance can be less than 30'X). ahout half a stop to a photographer.
SOLAR RADIATION AT THE GROUND As a consequence of attenuation. radiation has two distinct directional properties when it reaches the ground. Direct radiation arrives from the direction of the solar disc and includes a small component scattered directly forward. The term diffuse describes all other scattered radiation received from the blue sky (including the very bright aureole surrounding the sun) and from clouds, either by reflection or by transmission. The sum of the energy
41
flux densities for direct and diffuse radiation is known as lOtal or global radiation and for climatological purposes is measured on a horizontal surface.
Direct radiation Direct radiation at the ground, measured at right angles to the beam, rarely exceeds 75% of the Solar Constant, i.e. about 1030 W m -2 The minimum loss of 25% is attributable to molecular scattering and to absorption in almost equal proportions. Generally. aerosol is responsible for significant attenuation which increases the ratio of diffuse to total radiation and also changes spectral composition. Several forms of turbidity coefficient are used to quantify attenuation by aerosol. One of the most straightforward follows from applying Beer's Law to the atmosphere. If the irradiance of the direct beam below an aerosol-free atmosphere is Sp(O), radiation in the presence of aerosol can be written (4.4) where T is a turhidity coefficient and m is air mass. The value of T can be determined hy measuring Sp(T) as a function of In = sec tV and by calculating Sp(O) (also a function of m) from the properties of a clean atmosphere containing specified amounts of absorbing and scattering gases. A series of measurements in Britain gave values of T ranging from 0.07 in very clean air at the top of Ben Nevis to 0.6 beneath very polluted air in the English Midlands (Unsworth and Monteith, 1972). Corresponding values of exp( -Ttn) for IjJ = 30° are 0.92 and 0.50 implying losses of up to 50% of radiant energy. The spectrum of direct radiation depends strongly on the pathkngth of the beam and therefore on solar zenith angle. Figure 4.2 shows that the spectral
I.
II n "1
:, 1, . •
,
E 800
oJ..
!•,~
,,
~
•
~ ~
N
'E
,,
"
::; c 400
• u •
Sb
~
~
I
03
0.4
0 5
0.6
07
08
09
I
10
1.1
12
13
14
Wavelength (f./-m)
Fig. 4.2 Spectral distribution of total solar radiation (upper curve) and direct solar radiation (lower curve) calculated for a model atmosphere by Tooming and Gulyaev (1967). Solar elevation is 300 (m = 2) and precipitable water is 21 mm. The shaded area represents the diffuse flux which has maximum energy per unit wavelength at about 0.46 IJ-m.
42
Solar Radiation at the Ground
Radiation Environment
irradiance calculated for the solar beam is almost constant between 500 and 700 nm whereas, in the corresponding black-body spectrum, irradiance decreases markedly as wavelength increases beyond Am = 5m nm (Fig. 3.5). The difference is mainly a consequence of Rayleigh scattering which changes with wavelength in the opposite direction from the black-body spectrum; and ozune absorption is implicated too. As zenith angk increases, attenuation by scattering becomes very pronounced and the wavelength for maximum irradiance moves into the infra-red waveband when the sun is less than 20° above the horizon. The measurements referred to above also showed that for zenith angles between 40° and 60°, the ratio of visible to total radiation in the direct beam decreased from a maximum of about 0.5 in clean air to about 0.4 in very turbid air. The maximum exceeds the figure of 0.45 for extraterrestrial radiation because losses of visible radiation by scattering are more than offset by losses of infra-red radiation absorbed by water vapour and oxygen (Fig. 4.1). The quantum content of direct radiation increased with turbidity from a minimum of about 2.7 j.Lmol J- I total radiation in clean air to about 2.8 !L mol rl in turbid air. Few comparable figures are available for other sites but Weiss and Norman (1985) in Nebraska reported that the visible: total energy ratio was close to 0.46 for all sky conditions and zenith angles up to 800. (Confusion has arisen in the literature where these values for direct radiation are compared with values referring to total radiation as given later.)
s~atte:ed
43
radiation is. attenuated by an air mass depending on its erceived dlfectIon so. the horizon appear relatively de To thiS ;anahon, ambitiOUS architects and pedantic professgrs describe bye ra lance Istnbutlon of overcast skies as a function of zenith angle given
~~Iow ~or
a~d
r~glOns ne.a~ N(IjJ)
=
l~ted).
N(O)(I +bcosljJ)/(1 +b)
defines 'd th This t th distribution b ( .a Standard Overcast Sky . M easurements 10'Icate h a h e num er 1 + b) which is the ratio of radiance at the zenith to that at t e omon IS typically III the range 2.1-2.4 (Steven and Unsworth, 1979) values supported by theoretIcal analysis (Goudriaan, 1977) which also show~ the dependence on surface reHechon coefficients. The value (1 + b) = 3 in cthomdmfofn us.e, ISd.based on photometric studies and significantly overestim~tes e I use Irra lance of surfaces. Under a cloudless sky, the angular distribution of skylight depends on the posilion of the sun and cannot be described by any simple relation. In
,
i,,
,i
,,
0-8- )
, ,'
,I I'
I
1
DiHuse radiation
I
Beneath a clean, cloudless atmosphere, the absolute amount of diffuse radiation increases to a maximum somewhat less than 200 W m - 2 when l\I is less than 500 and the ratio of diffuse to total radiation falls between 0.1 and 0.15. As turbidity increases, so does Sd/S, and for ~I < 6(J°, observations in the
,
English Midlands fit the relation
S,,1S, = (J.bST + 0.10
(4.5) )-+-+--t-
For IjJ > 60 SiS, is a function of IjJ and is larger than equation (4.5) predicts. With increasing cloud amount also, Sd/St increases and reaches unity when the sun is obscured by dense cloud: but the absolute level of Sd is maximal when cloud cover is about 50%. The spectral composition of diffuse radiation is also strongly influenced by cloudiness. Beneath a cloudless sky, energy is predominantly within the visible spectrum (Fig. 4.2) but as cloud increases the ratio decreases towards the value of 0.5 characteristic of total radiation.
_+--f-+--+----"
O
30
60
90
0 ,
~--O. 5 - - - " " "
,,.i ,Ii
I
} ,I , 'I
",
I'
I
Angular distribution of skylight Under an overcast sky, the flux of solar radiation' received at the ground is almost completely diffuse. If it were perfectly diffuse, the radiance of the cloud base observed from the ground would be uniform and would therefore be equal to Suh ' from equation (3.23). The source providing this distribution is known as a Uniform Overcast Sky (UOS). In practice, the average radiance of a heavily overcast sky is hetween two and three times greater at the zenith than the horizon (because multiple-
,
180
,•
0
Stand?rd distribution for sO.lar zenith angle 35 of normalized sky radiance ~N/Sd wh~re N IS the v~lue .of sky radiance at a point and 'JrN is the diffuse flux which t e surface would receIve If the whole sky were uniformly bright (see p 42) (from . Steven, 1977). Fig. 4.3
Solar Radiation at the Ground 44
4~
Radiation Environment
general, the sky round the sun iS,much brighte~ tha? elsewhere b~cause there is a preponderance of scattering In a ~orwar,d duectlO!l bU,t there IS a sector of sky about 90" from the sun where the IntensIty of skylIght IS below the average for the hemisphere (Fig. 4.3). On average, the dIffuse radIatIon from a blue sky tends to be stronger nearer the horizon thall: at the zenIth, A~ the atmosphere becomes more dusty, the general effect IS to reduce the radiance of the circumsolar region and to increase the relative radiance of the upper part of the sky at the expense of regions near the horizon. Consequently the angular distribution of radiance becomes more umform as turbidity JOcreases,
1000
25.661
800 -
•
,
600
N
0
E
"
m'
1'-;
00
N
,
~
,
o N
"c .• 0
"'-
Net Radiation
56 Radiation Environment positively correlated with sunshine. In winter the correlation was negativ~: sunny cloudless days were days of minimum net radiation when the mean aIr temperature was below average. Staff at the University of Bergen have maintained records of incoming
600 .
400
,
-
N
,.
.: Lu
L,
E
•0 200 -
--
L,
L,
c
~
"0
~
5,
St
0 -100
Re
Re
03
00
06
12
09
15
18
21
24
57
short- and long-wave radiation for many years and have overcome the problem of maintaining a standard surface by using black-body radiation at air (i.e. screen) temperature for Lu and by adopting a reflection coefficient of 0.2 (grass) or 0.7 (snow) as appropriate. The shaded part of Fig. 4.12 represents the difference between the fluxes I~d and Lu, i.e. the net loss of long-wave radiation; the bold line is Rn. In cloudless weather, the diurnal change in the two long-wave components is much smaller than the change of short-wave radiation which follows an almost sinusoidal curve. The curve for net radiation is therefore almost parallel to the S, curve during the day, decreases to a minimum value in the early evening and then increases very slowly for the rest of the night (because the lower atmosphere is cooled by radiative exchange with the earth's surface). In summer, the period during which Rn is positive is usually about 2 or 3 hours shorter than the period during which SI is positive. Comparison of the curves for clear spring and winter days (Figs. 4.12a and b) shows that the seasonal change of SiRn (Fig. 4.11) is a consequence of (i) the shorter period of daylight in winter and (ii) much smaller maximum values of SI in winter, unmatched by an equivalent decrease in_ the net long-wave loss. In overcast weather, Ld becomes almost equal to uTa4 ; Rn is almost zero at night, and during the day Rn = (1- p)S,. The device of referring net radiation to a surface at air temperature is convenient in microdimatology too (see p. 187) and deserves to be more widely adopted.
,i
I •
Local time i,
,
!, , ,1'I·
(al
..
I .,~
. I,·
I
400
,
N
,. • -.-
L,
E
~
i ,,
L, .
200
L, "/
•
.....
L, ,
I .,
c
"0
'"
0 -100
i
5,/ "'-
~
~
,
·,
Re .............
00
03
·'
•
Re
~
06
09
15
12
18
21
'
.
24
Local time \
(bl 0
Fig.4.12 Radiation balance at Bergen, Norway (60 N, SOE): (a) on 13 April 1968, (b) on 11 January 1968. The grey area shows the net long wave loss and the ~ine R? is net radiation. Note that net radiation was calculated from measured fluxes of mcommg short and long wave radiation, assuming that the reflectivity of the surface was 0.20 in April (e.g. vegetation) and 0.70 in January (e.g. snow). The radiative temperature of the surface was assumed equal to the measured air temperature.
....
Direct Solar Radiation
59
MICROCLIMATOLOGY OF RADIATION (i) Interception
, I
In conventional problems of micrometeorology, fluxes of radiation at the earth's surface are measured and specified by the receipt or loss of energy per unit area of a horizontal plane. To estimate the amount of radiation intercepted by the surface of a plant or animal, the horizontal irradiance must be multiplied by a shape factor depending on (i) the geometry of the surface and (ii) the directional properties of the radiation. To make the analysis more manageable, objects such as spheres or cylinders with a relatively simple geometry are often used to represent the more irregular shapes of plants and animals. Appropriate shape factors for these models will be derived. The radiation intercepted by an organism Of its analogue can be expressed as the mean flux per unit area of surface. A bar will be used to distinguish this measure of irradiance from the conventional flux on a horizontal surface, e.g. when solar irradiance is S (W per m2 horizontal area), the corresponding 2 irradiance of a sheep or a cylinder will be written S (W per m total area).
"", ,
"",, \
J.'igo. 5.1 Area A projected on surface at right angles to solar beam (A ) and on P honzontal surface (Ah). ,
.
"
.,,
Shape fadors
,
.
•
Sphere
SOLAR RADIATION The flux of direct solar radiation is usually measured either on a horizontal plane (So) or on a plane perpendicular to the sun's rays (Sr)' For any solid object exposed to direct radiation at e1evation ~, a simple relation between the mean flux Sb and the horizontal flux Sb can be derived from the relation between the area of shadow A h cast on a horizontal surface and the area projected in the direction of the beam Ap. The area projected in the direction of the sun's rays is Ap = Ah sin ~ (Fig. 5.1) and the intercepted flux is (5.1 ) i.e. the area of shadow on a horizontal plane times the irradiance of that plane. Then if the surface area of the object is A (5.2)
The shape factor (Ao/A) can be calculated from geometrical principles or measured directly from the area of a shadow or from a shadow photograph.
",
The shadow cast by a sphere of radius r has an area of "r/sin The surface area of the sphere is 4"r so
"r A = 4"r sin 13 = 0.25 cosec 13
Ah
13 (Fig. 5.2).
I,
(5.3)
The mean irradiance of a sphere is therefore Sh = (0.25 cosec
13) Sb = 0.25Sp
(5.4)
Ellipsoid ~
spher: is a. special t~pe of ellipsoid-a solid whose cross-section is elliptical m one directIOn and clfcular about an axis of rotation at right-angles to that direct~on. If t~c e~liptical cross-~ection through the centre of the ellipsoid has a vertical semi-aXIS a and a honzontal semi-axis b (Fig. 5.3), it can be shown that a beam at elevation will cast a horizontal shadow with area Ah = "b 2 {1 + a2/(b 2 tan 2 j3)}tl5 (5.5)
a
, ,!
, , ,,
61
Direct Solar Radiation 60
Microclimatology of Radiation which reduces to Ah
=
'ITb 2 for a vertical beam and to Ah
= TIb
2
/sin rJ when
a = b.
The surface area of an oblate spheroid (b > a) is A = 2'lTb 2 [1 + (a 2/2b 2 E I)
(5.6)
where
(5.7)
(1- a'/b')'"
EI =
and the area of a prolate spheroid (a> b) is
A = 21Tb 2{1 + (a/bE,) sin-I E,}
(5.8)
where E2
(5.9)
b'/a')"5
= (I -
The ratio AliA can now be found by dividing equation (5.5) by equation (5.6) or (5.8) as appropriate (Fig. 5.3). Inclined plane
Figure 5.4 shows the end and plan views of a square plane with sides of unit length making an angle a with the horizon XY and exposed to a beam of ,
Isin.8
.
,I,, "
Fig. 5.2
Geometry of sphere projected on horizontal surface.
1
I
.
- '-':,'.-,' --- " " - " ' --",",', ". -- ---",' , " -" " ' , " .'.-,-.';'.'.-' ',-, ':-. .', -'" ',- - '-','," '.-, ,--
'.- --
sin ,
a
x - - - - . ~~'·.iC·".iC x (not shown), the walls are fully illuminated and the shadow cast on a horizontal surface by the whole cone is simply the shadow area of the base Ah = 'IT cos2 u. As the area of the walls is A = 1T cos (l, the shape factorfor the walls alone is A hi A = cos Ci. When fl < (l, the shadow has a more complex form. The walls are now partly in shadow: at the base, COB is illuminated, BEC is not, and the shadow can be delineated by projecting tangents at Band C to meet at A. The sector of the cone in shadow can now be specified by the angle AOB = AOC = 00 , Now
OB
.;
•
I
;
.a
All leaves are illuminated from above, i.e. on their upper (adaxial) surfaces. The whole curved surface of the cone is illuminated so
,
Ah
11'
')[.=-= s
A
cos- 0.
Some of the leaves are illuminated from below, i.e. on their lower (abaxial) surfaces. In accordance with the method used for the sphere and the cylinder, the calle can be split into two parts. The relative shadow area of the convex part representing the abaxial surfaces has been calculated already (p. 68): it is cos u{{ 1T - 80 ) cos a + sin 0. cot 13 sin 80 }. The shadow area of the concave part (adaxial surfaces) is the area ACEB in Fig. 5.12, i.e. the sum of two triangles less the sector of the circle or cos u{sin a cot 13 sin So - So cos u}. The total shadow area is the sum of these expressions and as A = "IT cos a for the curved surface alone,
where
~I
\%9) (a)
Idealized leaf distributions
'j(
90
,,=
(b)
0.00 0.50 0.50 0.87
solar elevation [3 60 30
0.37 0.58 0.50 0.87
I.lO 1.00 0.58 0.87
Real canopies
.II, 1.10 0.97
0.86 0.94 0.70 0.69 0.63
0.49 0.43 0.29 0.20
11'COSU
~ 25°. It also 1~c1udes ra?la~lOn ,from a,uniform overcast sky which is transmitted by a sphencal leaf dIstnbutlOn as If all the radiation emanated from an angle of about 40°. For these restricted categories of foliage, the value of Xb to be inserted in equation (6.4) is given by the relative shadow area of the assembly X, as evaluated in Chapter 5. After integration of the equations (3.27, 3.28) which describe the downward and upward streams of radiation in a canopy, the reflection coefficient is given by Pc =
(6.3)
The quantities U o Pc and Tc can be derived from the Kubclka-Munk equations as functions of (i) the corresponding coefficients for leaves; (ii) the leaf area index; (iii) a canopy attenuation coefficient (6.4) (from equation (3.32» where X h is the coefficient for black leaves (u ~ I) with the same geometry. Strictly, the theory can be applied only to stands with horizontal leaves so that Xb = 1. This ensures that the intercepting area of all leaves is independent of the direction of incident radiation and therefore a single value of 'X is
Pc' + fexp (-n{L)
I +pc'fexp (-2XL) and the transmission coefficient by
(6.5)
_ {(IV 2 - I )/(p,p/ - I)} exp (-XL) Tc 1+ pc'fexp (-2XL)
(6.6)
f~(pc'-pJ/(Pc'p,-I)
(6.7)
where Neglecting the second-order terms Pc *2 and Pc *p~ gives approximate values of the coefficients as Pc ~ Pc' - (Pc' - p,) exp (-2XL)
(6.8)
and T,
~
exp (-J{L)
(6.9)
where Pc clearly has limits of Pc * when L is large and Ps when L is small. Now, from equations (6.8) and (6.9), absorption by the canopy is U
c = 1- {p,' - (p,' - p,) exp (-2XL)) - (I - p,) exp (-:J{L)
(6.10)
The ten~ (p,' - p,) exp (-2~{L) will be much smaller than (1 - p,) exp (-XL) when p, IS small andlor ~{L IS large. The canopy absorptIOn then becomes Uc
l+TcPs~Tc~Pc
89
= 1 ~ Pc * ~ (1 - Ps)Tc = (1- p, *)(I-TJ +T,(p, -
Pc ')
(6.11 )
Some practical implications of these relations will now be considered.
Absorbed and intercepted radiation In the, field, the rad!ati~n intercepted by a crop stand is conveniently determmed by mountmg IOstruments such as tube solarimeters above and below ~h~ canopy. Most tube solarimeters in common use respond uniformly to radIation through the solar spectrum. The fraction of the incident flux d.ensity recorded by the lower solarimeter is Tc and the intercepted fraction is SImply 1- T,.
90
Microclimatology of Radiation
Radiative Properties of Natural Materials
Empirically. the amount of total solar radiation intercepted by a canopy is often well correlated with the production of dry matter during periods when the leaf area index is increasing (Russell et al .• 1989). Theoretically, however, growth rate should depend on the absorbed fraction of radiation. Equation (6.11) provides reassurance that, subject to the validity of the appn)XJmatlOns used, and provided the term 7~.:CP~ - Pc *) is small, U c is a nearly constant fraction of (1- T,). The fraction is 1- Pc', i.e. about 0.94 for the visible spectrum and 0.77 for total solar radiation using the values derived above. A further step is needed to obtain absorbed PAR from intercepted total
radiation. From equations (6.4) and (6.9) (6.12) The transmission of PAR, 70,;1>, can therefore be estimated from the transmission of total radiation, Ta, using TeP
= Ta exp {("TO 5 -
"pO.5)X b L}
= Tcr exp (-0.26JfbL)
if "T = 004 and "p equation (6.11) is
(6.13)
= 0.8. The fraction of PAR absorbed by a stand as given by ",p = O.94{I -
ToT
exp (-O.26XbL)}
(6.14)
assuming p,p' = 0.06 and neglecting the small term T,p(p,p - PcP '). Finally, the fractional absorption of energy (PAR) can be converted to the equivalent fraction of absorbed quanta using a coefficient of 4.6 flmole per joule (p. 49). Figure 6.7 shows how absorbed quanta derived from equation (6.14) are related to intercepted total radiation from equation (6.9). Equation (6.9) can also be written Ta =
exp (-JC' L)
a U.5X b is the attentuation coefficient a conventionally determined by plotting In ToT against L to find -J(, as a slope. where 'X'
=
1.0
0.6
08
"j: 0.8 ~ 1. 0.6
-
o
0..
u
acp /
0.4
a
/
0.2
0.5
1
2
4
I
I
I
0.5
1
2
4
/
/
I
I
I
0.5
1
/
/
/
The ratio of attenuation coefficients for PAR and total radiation is (",J"T )05, a conservative quantity because the fraction of total radiation absorbed by a leaf is largely determined by pigments which absorb in the PAR waveband. To demonstrate this point, assume that in the PAR waveband pp = Tp = 0.05 so that "P = 0.9, that absorptivity in the infra-red waveband is 0.1 and that the solar spectrum contains equal amounts of energy in the two wavebands (p. 48). The absorgtivity for total radi(ltirm ie "T = (0.9+0.1)/2 = 0.5 and Xp/XT = (0.9/0.5) .5 = 1.34. Doubling PI and Tp
to 0.1 makes "p = 0.8, "T = 0.45 and Xpf.J{T = (0.8/0045)°·5 = 1.33. Green (1984) found that the ratio of X for quanta (effectively the same as for PAR) and for total radiation was 1.34 for wheat grown with a range of nitrogen applications so that there was a conspicuous range of greenness between treatments.
Remote sensing Because the spectrum of radiation reHected by foliage has a different shape from the spectrum for all types of soil, the extent to which soil is covered by vegetation can be estimated from the reflection spectrum of the area, as recorded from an aircraft or a satellite. However, interpretation can be very
difficult if the cover is not uniform. Most information is obtained by working near the 'red edge' at 700 nm below which leaves are almost black and above which almost all the incident radiation is scattered. In principle, an estimate of cover could be obtained simply from the reflectivity Pi in the near infra-red
(say between 700 and 900 nm), but in practice, it is often difficult to measure the incident and reflected fluxes simultaneously and error is unavoidable if the
incident Hux is changing with time. The difficulty is usually overcome by measuring the ratio x I of the absolute amounts of radiation reflected in the near infra-red and in some part of the Visible spectrum, usually in the red. The corresponding ratio x, for the spectrum of incident radiation (which changes much more slowly than the absolute flux at any wavelength) can be obtained by recording the spectrum reHected from a standard white surface or by calculation if the measurements are above an atmosphere whose radiative behaviour is known. Then the required ratio of infra and red reflectivities is
/'
given by
I 11- TCT
2
91
4
8
Pi
reflected IRiincident IR
p,
reflected Rlincident R reHected I Rlreflected R
xI
incident IRiincident R
x,
(6.15)
Many workers use a 'normalized difference vegetation index' (NOVI) defined as
Leaf area index
Fig. 6.7 Fractional absorption of visible radiation (or PAR) O'.CP (full lines), and fractional interception of total radiation (1- TcT) as functions of leaf area index for three values of the attenuation coefficient for black leaves XI> (assuming pp ~ Tp ~ 0.05).
Pi-Pr
XI-Xz
Pi+Pr
Xt+ X 2
(6.16)
The correlation between fractional ground cover and this quantity is seldom
92
Radiative Properties of Natural Materials
Microclimatology of Radiation
much better than the correlation with the simple ratio ~1/X2 but the fact that the ratio has limits of - 1 and + 1 makes It convement to handle when computing. . ' . The main problems in using both types of ~ndex to ~stImate umform ground cover are (i) changes in the spectral propertIes of fohage as a consequence of senescence, poor nutrition or disease which alter the dependence of the mdex on leaf area (Steven, 1983); and (ii) changes in the spectrum of reflected radiation between the ground and a detector as a consequence of scattenng In the atmosphere (Steven et al., 1984). Nevertheless, Tucker et al. (1985) were able to obtain plausible distributions of ground cover month by month over the whole African continent; and by integration they established a seasonal index of total intercepted radiation and therefore of bioma~s product~on. A similar exercise over the North American continent gave blO~ass estImat~s consistent with accepted figures for regional classes of vegetation. The basIs for this procedure is now outlined. . . Following the example of Asrar et at. (1984) but usmg the approXImate equations already derived, a relation can be established bet~ee.n the obs~rved value of P/Pr for a canopy and the fraction of P ~R which It transmits or absorbs. First, the attenuation coefficient can be wflttcn ".lL -_ aj o.,"{ (6.17) .J h J
where j is replaced by P, r or ; for the wavebands of PAR, red or near infra-red radiation.
-
Then from equations (6.8) and (6.9) and with the same conventions for subscripts
p, p,
(2X,!'Xr) ~I Tp Pl"l *-(p CI *-p.) )T PX,rx,) Pcr * - (pcr * - PsrP
(6.18)
It follows that, within the limits of the approximations used, the ratio P/Pr should be a unique function of Tp, independent of Xb and therefore of leaf architecture because it involves the ratio of coefficients at different wavelengths. The function also depends on the values of aj, a r and ap which determine the exponents of Tp and the soil reflectivities P~i and Psr' When the second term in the denominator is small compared with the first term, P/Pr is a function of Tp(2.H,/·.I{I'). For the special case a/ar = 114 (e.g. Ui = 0.2, Ct, = 0.8), ~(if:){P = v174 = 1/2 and pip, is a linear function of Tp. Figure 6.8 shows the extent to which the function departs from linearity because the approximations behind this result are not valid over the whole range of Tp. The spectral ratio P/Pr is therefore a valid and extremely useful index of the radiation illtercepted by and therefore the radiation absorbed by a canopy. It can legitimately be used to provide an estimate of growth rate when the relation between growth and intercepted radiation is known. In contrast, the relation hetween P/Pr and either biomass or leaf area is strongly non-linear and depends on leaf architecture (Fig. 6.7).
Animals
15
o
93
10
0.8
5 0.6
o~--~~----~----c 02 0.4 0.6
0.8
10
Fractional absorption of PAR
Fig. 6.8 Relation between far-red: red reflectivity ratio for a canopy of vegetation and the amount of PAR absorbed by the canopy. Leaf absorptivity specified.
The coats and skins of animals reflect solar radiation both in the visible and in the infra-red regions of the spectrum, and reflectivity is a function of the angle of incidence of the radiation, as for ,,,'ater and leaves. Figure 6.9 shows that the reflectivity of hair coats increases throughout the visible spectrum (like soil); and that intra- and inter-specific differences of reflectivity are much larger than for leaves. The maximum reflectivity of several types of coat is found between I amI 2 flm and water is responsible for absorption at 1.45 and 1.95 flm. Beyond 3 flm, the absorptivity and emissivity of most types of coat is between 90 and 95%. The measurements of coat reflectivity plotted in Fig. 6.9 were made in the laboratory on samples of coats exposed to radiation at normal incidence. When Hutchinson et al. (1975) used a miniature solarimetcr to measure reflection from the coats of Jive animals standing in the field, they found a marked dependence of reflection coefficient on the local angle of incidence of direct sunlight. For Jersey heifers with relatively smooth coats exposed to sun at an elevation of 48" (Fig. 6.9), a minimum reflection coefficient of 0.25 was recorded when the angle of incidence was minimal but p was more than 0.7 at grazing incidence because of the large contribution from specular reflection. The change of reflection with angle was much smaller for Merino sheep with deep, woolly coats (Fig. 6.10). The implication of this work is that laboratory measurements of reflection, usually made at normal incidence, may be an unreliable guide to the effective reflection coefficient for an animal in the field, particularly for species with relatively smooth coats. Table 6.1 summa-
I
Radiative Properties of Natural Materials
94 Microclimatotogy of Radiation
100
I
:.
( a) Buffalo
I
I
I
Approximate > visible spectrum
I I
__------~~......._
0.6
'!....-~".
80
White rot
I I
I
Holstein, white
-
I
.> .00.4
I
CD
u
I
--
560
Jersey, ton
u
0.5
Q)
....-
CD CD
Q)
~
0::
o
4
(6.22)
where the bars indicate averaging over the exposed surface. The sheep is assumed to be a horizontal cylinder with its axis at right angles to the sun's rays, reflectivity is P3 and mean surface temperature is T 3 . A man For a man standing on the lawn, the radiation balance is formally identical to equation (6.22) with P4, T4 replacing P3, T 3 ·
(a) During the day, the lawn absorbs more net radiation than any of the other surfaces including the isolated leaf. The leaf receives less short-wave radiation than the lawn + p,)S,I2 compared with S,) and absorbs more long-wave radiation «La + crT,4)/2 compared with La). (b) The sheep absorbs less net radiation than the other surfaces. This is partly a consequence of the relatively large reflection coefficient (0.4) and partly a consequence of geometry. (c) The geometry of the man ensures that Rn is large in relation to other surfaces when the sun is low . (d) For all surfaces, the net radiation is greatest when the sun is shining between clouds and is larger under an overcast sky than it is when the sun is near the horizon.
«I
~
-
Values assumed for the radiation fluxes and for p and T are given in Table 6.2. Long-wave emissivities are assumed to be unity. It is instructive to compare the net radiation received by different surfaces at the same time, noting the effects of geometry and reflection coefficients; or to compare the same surface at different times to see the influence of solar elevation and cloudiness. Salient features of these comparisons are as follows.
I
-• •
-
R"~(I-P3)(I+p,)S,+Ld+Le-crT3
."
.-
where Pl, T2 are the reflection coefficient and radiative temperature of the leaf. Note that R" is the net radiation per unit of total leaf area, i.e. twice the plane area or twice the leaf area index. A sheep The sheep is assumed to be standing on the lawn and so receives radiation reflected and emitted by the surface below it. When the area of shadow is ignored the net radiation for the sheep is
...-.., .-'" • •
""
,, :!
.',·,I I I
" q I, • •
•
•
, i •
, •
100
Microclimatology of Radiation
Table 6.2
Conditions assumed for radiation balances, Fig. 6.12 I
2
3
4
5
High
High sun partly cloudy
Low sun clear
Overcast day
Clear
sun clear Solar elevation ~ Direct solar radiation Sb(Wm- 2 ) Diffuse solar radiation Sd (Wm-') Downward long wave radiation Ld (W m- 2 )
night
60
60
10
800
800
80
100
250
30
250
320
370
310
380
270
20 24 24 33 38
20 24 25 36 39
18 15 15 15 15
15 15 15 20 20
10
Surface temperature (OC) •
alf
lawn leaf
sheep man Reflectivities
lawn leaf sheep man
0.23 0.25 0.40 0.15
0.23 0.25 0.40 0.15
0.25 0.35 0.40 0.15
6
4 10 10
0.23 0.25 0040 0.15
(e) At night, the leaf, sheep and man receive long-wave radiation from the lawn as well as from the sky so their net loss of long-wave radiation is less than the net loss from the lawn.
Measurement Measuring the net exchange of radiation at the surface of a uniform plane poses no problems: a net radiometer is exposed with the surface of its thermopile parallel to the plane. For more complex surfaces, the net flux can be derived from an application of Green's theorem which states that the flux of any quantity received within or lost by a defined element of space is the integral of the flux evaluated at right angles to the envelope which defines the space. This principle has been applied to measure the net radiant exchange of apple trees with tubular net radiometers defining the surface of a cylinder and with thermopile surfaces parallel to the surface of the cylinder (Thorpe, 1978). Similarly, Funk (1964) measured the net radiant flux received by a man standing inside a vertical cylindrical framework over which a single net radiometer moved on spiral guides, always pointing towards the axis of the cylinder.
MOMENTUM TRANSFER
When plants or animals are exposed to radiation, the energy which they absorb can be used in three ways: for heating, for the evaporation of water, and for photochemical reactions. Heating of the organism itself or of its environment implies a transfer of heat by conduction or by convection; evaporation involves a transfer of water vapour molecules in the system and photosynthesis involves a similar transfer of carbon dioxide molecules. At the surface of an organism, heat and mass transfer are sustained by molecular diffusion through a thin skin of air known as a boundary layer in contact with the surface. The behaviour of this layer depends on the viscous properties of air and on the transfer of momentum associated with viscous forces. A discussion uf momentum transfer is therefore needed as background to the following three chapters which consider different aspects of exchange between organisms and their environment.
BOUNDARY LAYERS
,
~
,,
,,
Figure 7.1 shows the development of a boundary layer over a smooth flat surface immersed in a moving fluid (i.e. a gas Or liquid). When the streamlines of flow are almost parallel to the surface the layer is said to be laminar and the flow of momentum across it is affected by the momentum exchange between individual molecules discussed on p. 18. The thickness of a laminar boundary layer cannot increase indefinitely because the flow becomes unstable and breaks down to a chaotic pattern of swirling motions called a turbulent boundary layer. A second laminar layer of restricted depth-the laminar sublayer-forms immediately above the surface and below the turbulent layer. The transition from laminar to turbulent flow depends on the relative size of inertial forces associated with the horizontal movement of the fluid to viscous forces generated by inter-molecular attraction, sometimes referred to as 'internal friction'. The ratio of inertial to viscous forces is known as the Reynolds number (Re) after the physicist who first showed that this ratio determined the onset of turbulence when a liquid flowed through a pipe. When Re is small, viscous forces predominate so that the flow tends to remain laminar but when the ratio increases beyond a critical value Re e , inertial forces take charge and the flow becomes turbulent.
,
.~
102
BOllndary Layers
Momentum Transfer £
Velocity profiles (a)
Uniform air stream
103
( b)
(el
Laminar boundary layer
Turbulent boundary layer
c Unmodified flow veloCity V
Boundary layer velocity
~
-
o-
I,
j
I
60
,,
0
E
~
•
'" 40
o
'" ~ o
I
0:: ~
•
0
-'"o
I
20
-,_L,------00------:-'------~2:-----3-------~----~--~~ 4 5 6 log (Re)
Fig. 7.6 Relation between the drag coefficient for a sphere and Reynolds number. The pecked line is based on Stokes' Law. The discontinuity at Re = 3 x 105 corresponds to the transition from a laminar to a turbulent boundary layer (Mercer, 1973).
o
1/'
, .... 0
°O~--~--o,--~I~o~~ILI----~I____-L____~I~__-L____~)
5
10
15
20
Windspeed (m s-'l Fig. 7.7 Percentage of spores removed by blowing for 15 s on spores reared on dried plant material (from Aylor, 1975). Vertical bars represent standard deviations.
112
Momentum Transfer Wind Profiles and Drag on Uniform Surfaces
Rep = IO (similar to the value for a sphere, Fig. 7.6), yields Fm = 1 X 10- 7 N. Aylor also used a centrifugal method of detachIng spores, and thIs gave
113
and
excellent agreement with his est~m.ates of Fm- ~po~es .are det~ched at much lower mean windspeeds than this In the field, tndlcatmg the Importance of
duldz = u*/(kz) T
brief gusts in breaking down the leaf boundary layer and dispersing pathogens
into the atmosphere.
I
(7.15b)
= pu* 2
(7.15c)
Integration of equation (7.15b) yields
u
PROFILES AND DRAG ON UNIFORM SURFACES
where Z
In Chapter 3, it was shown that, the vertical flux, ~)f a~ en.lity ~' can be represented by the. mean v~lue ~1 the product pw. s which. ~s fiflltc when fluctuations of vertICal velocity ware correlated, either posItively or negatively, with simultaneous fluctuations of the. entity Sf. W.hen the cnt~ty i~ horizontal momentum, s' becomes the fluctuation of the hOflzontai velocity u and the vertical flux of horizontal momentum is given by pu' w' . Provided
T
= pu w = pu*
is a constant termed the roughness length such that u = 0 when
Zoo
surface elements by assuming the existence of a zero plane at a height d such that the distribution of shearing stress over the elements is aerodynamically equivalent to the imposition~ of the entire stress at height d. The wind profile equation then becomes
(7.10)
u = (u,lk) In {(z-d)IzII }
where U*, known as the friction velocity, is given by
(7.11 )
Short
grass
Tall crop
1
4
/
(7.12)
3
--
biz
(7.13)
•
•"
(7.12) and (7.13) in equation (7.11) gives
I
I -
(7.14)
\
,
equation (7.14) implies that
a=ku*;b=u*lk
1
2
3
,
,, (
i/\
.
,
1-
'/\
'( ,rI / ' 1
I
d
2
3
Windspeed (m 5- 1)
where k is a constant. Substitution in equations (7.12) to (7.14) now gives (7.15a)
I
'I,
I
oLLl~LL'L ~'--'_\U-'-!f_LL LU~J~ 0 \
o
. ( .
i
,
,
I
,
~
~/
implying that ab = u* 2. Furthermore, because a and b are both velocities,
Km = ku*z
1
"
./
•
2
where b is a second constant with dimensions of velocity. Putting equations
T=pab
Ii
3
.I
E
~
=
4
jI
where a is a quantity with the dimensions of velocity. Similarly, the simplest assumption for the wind gradient is
duldz
'
,
(7.18)
functional relations. The simplest assumption for the height dependence of K is
az
, ,
These equations are valid in the air above the surface elements where the assumption that shearing stress is independent of height is satisfied. Because
where KM is a turbulent transfer coefficient for momentu~ .wit~ dimensi~ns 2 1 L T- . From experience, both windspeed and !urbulcnt mlxmg mcreas~ with height, and a simple dimensional argument will now he used to obtam the
KM =
I ,"
KM = ku*(z-d)
The shearing stress may also be written as = pKMduldz
(7.17)
and equation (7.15a) becomes
u* = (U,2)O.5 = (W,')05
T
(7.16)
A more general form of equation (7.16) takes account of the height of the
force per unit ground area, otherwise known as the shearing stress ~T). Moreover, if the intensity of turbulence is the same in horizontal and vertical directions, u' = w' so that 2
(u*lk) In (Zlzll)
This is the logarithmic wind-profile equation, found to be valid over many types of uniform surface provided the conditions listed on p. 233 are satisfied. Measurements confirm that k is a constant-the von Karman constant after a famous aerodynamicist-and it is usually assigned a value of 0.41, as determined by experiment.
the vertical flux is constant with height, this quantity can be identified as the
I,
=
Zo
=
Fig. 7.8 Wind profiles over short grass and a tall crop when the windspeed at 4 m above the ground is 5 m s - t. The lilled circles represent hypothetical measurements from sets of anemometers.
,
114
Wind Profiles and Drag on Uniform Surfaces
Momentum Transfer
115
crop respectively: T = 0.027 and 0.25 N m- 2 : and 0.25 and 0.58 m 2 s-' for the transfer coefficient KM at a height of 4 m. When windspeed is measured over heights much larger than d, this type of analysIs reqUires values UI and U2 at a minimum of two heights ZI and Z2 so both u* and Zo can be eliminated initially to give
of the absorption of momentum below the tops of the elements, the equation is not valid in this region, nor is it valid immediately above very tall vegetation where the relation between flux and gradient defined by equation (7.11) breaks down (see Chapter 14). It is nevertheless possible to use equation (7.17) to obtain an extrapolated value of u which tends to zero at a height given by z = Zo + d. To recap, d is an equivalent height for the absorption of momentum (a 'centre of pressure') and (d + zo) is an equivalent height for zero windspeed. . . To demonstrate the relation between wmdspeed and height for contrastmg surfaces, Fig. 7.8 contains profiles over short grass (d = 0.7 cm, Zo = 0.1 cm) and a tall crop (d = 95 cm, Zo = 20 cm) when the windspeed at 4 m is 3 m S-l Figure 7.9 shows the equivalent logarithmic plots of u as a function of In (z-d). Because height appears (by conventIOn) on the vert,cal aXIS, the slope is klu, giving values of u, = 0.15 and 0.46 m s··' for the grass and tall
Inzo=(u,lnz,-u,lnz2)/(u,-u,)
(7.19)
When the value of d is significant, at least three heights are needed so that
both u. and
Zo
can be eliminated initially to give U,-Il,
u, -u,
In(z,-d)-ln(z2-d) In (z, - d) -In (z, - d)
(7.20)
This equ~tion allows ~ to be found either by iteration in a computer program or graphIcally by ploltmg the right hand side as a function of d.
Behaviour of Zo and d 6·
The dependence of
Zo
and -d on vegetation height and structure has been
p
025 •
05 •
1
2
4
.-Gl
Zm/h
O~
.s::
cO.? IE Gl gO.6 I-
I
--l
0 0·0. O'J-
~ 1.1 for laminar flow (intensity of turbulence: i ~ 0.01 to (1.02) and for Rc lip tn 1 x 10-1 (\Viglc)' and Clark, 197-0. For turhulcnt flow (i = O.J to 0.4), A ~ (J.O-I and 1/ ~ O.H4, Nu exceeded the laminar flow value above (but
Measurements oJ Convection 130
131
Heat Transfer
10"
not below) Rc = and 13 was about 2.5 for Re = 10"Relatively little work has been published on free convection from leaves despite its significance for heat transfer in canopies where windspeeds below 0.5 m 5- 1 are common. In one study with leaves of Acer and Quercus, Nu was close to the prediction from equation (8.9) when GrPr was 10" but i> increased as GrPr decreased below this figure and was about 2 at GrPr = 10"The dissipation of heat from a set of copper plates exposed to low windspeed was studied by Vogel (1970). All the plates had the same area but their shapes ranged from a circle and a regular 6-point star to replicas of oak leaves with characteristic lobes (Fig. 8.4). The amount of electrical energy needed to keep each plate 15°C warmer than the surrounding air was recorded at windspeeds from 0 to 0.3 m S-1 and at different orientations. This energy is proportional to Nu and inversely proportional to the resistance 'H under the conditions of the experiment. The main conclusions were:
(i) (ii)
(iii) (iv) (v) (vi)
increasing airflow from 0 to n.3 m s-'\ decreased rH by 30 to 95%: the decrease was greater for the leaf models than for the stellate shapes; the resistances of all the stellatc and lobed platcs were smaller than the resistance of the round plate and were less sensitive to orientation; a deeply lobed model simulating a sun leaf of oak always had a smaller resistance than a shade leaf with smaller lobes; the resistance of the leaf models was least when the surface was oblique to the airstream; serrations about 5 mm deep on the periphery of the circular plate had no perceptible effect on its thermal resistance; the measurements could not be correlated using a simple Nusselt number based on a weighted mean width as described above.
These measurements support the hypothesis favoured b" some ecologists that the shapes of leaves may represent an adaptation to their thermal environment. Because the natural environment is very variable and because physiological responses to changes of leaf tissue temperatures are complex, conclusive experimental proof is still lacking. To recap, non-dimensional groups provide a useful way of summarizing and comparing heat loss from objects with similar geometry hut different size exposed to different wind regimes. Standard values of these groups should be regarded as a useful yardstick for estimating heat transfer in the field,
Su
Sh
Fig.8.4 Metal replicas of leaves used by Vogel (1970) to study heat losses in free convection. The shapes Su and Sh represent sun and shade leaves of white oak.
allowing mean surface temperature to he estimated within the precision needed for most ecological studies. When precision is essential, appropriate values of N u must be obtained experimentally. preferably by measurements on real leaves at the relevant site.
Cylinders and spheres The flow of air over cylindrical and spherical objects is more complex than the flow over plates because of the separation of the boundary layer that occurs towards the rear, and the wake generated by this separation. The Nusselt number can again be related to the Reynolds number by the expression Nu = ARe"Pro. J3 but, unlike the corresponding constants for flat plates, both A and n change wIth the value of Re (Table A.5(a». Both for cylinders and spheres, the obvious quantity to adopt for the characteristic dimension is the diameter but, for the irregular bodies of animals, the cube root of the volume may be more appropriate. Mammals McArthur measured the heat loss from a horizontal cylinder electrically heated and WIth a dIameter of 0.33 m to simulate the trunk of a sheep (McArthur and Monteith, 1980a). In a second set of measurements the ' cylinder was covered with the fleece of a sheep. ll For the bare cylinder, estimates of Nu fitted the relation Nu = A Re with 5 A = 0.095 and n = 0.68 in the region 2 X 104 < Re < 3 X 10 , ct. A = 0.17 and n = 0.62 in Table A.5(a). To avoid the difficulty of comparing equations with different exponents, FIg. 8.5 shows the reSIstance of cylInders and cylindrically-shaped animals obta~ned f~om var~ous sources. Consistency is reassuring, and as the aerodynamic resistance IS often small comp~,-red with the coat resistance of an animal, the degree of uncertainty implied by Fig. 8.5 is immaterial for most calculations. Nusselt numbers for hair surfaces are harder to establish because of the problem of measuring the temperature at the tips of the hairs. Using a radIatIOn thermometer to measure this quantity, McArthur found A = 0.112 and n = 0.88. Comparison of the relation between Nu and Re for smooth and hairy cylinders (Fig. 8.6) suggests that in a light wind «I m S--I) the effective boundary layer depth was thicker when the surface was composed of hairs, presumably because the effective surface 'seen' by the radiometer was some distance below the physical hair tips. But when Re exceeded lOS, the boundary layer depth was smaller with hair present. suggesting that wind penetrated the coat, possibly generating turbulence. When measurements of this kind are used to estimate heat loss from mammals, Nusselt numbers need to be obtained for appendages too: legs and tails can be treated as cylinders and heads as spheres. Rapp (1970) reviewed several sets of measurements of the loss of heat from nUd.e human subjects in different postures and exposed to a range of enVlfonments. Agreement between measured and standard Nusselt numbers was excellent except when the convection regime was mixed. In one
132
Measurements of Convection
Heat Transfer
4
133
3
d
3 ~
"
~
IE
Z
0
o
0'
-
VJ
~
,I
\
,
\
Q)
0
c
-.- 2
\
0
VJ VJ
Q)
a::
\
\
11.
\
,,
,,
o
' ..................
t)'-,
............ ,£ '-
1
2
.... ....
..............
--a _--- - --
5
4 log Re
1
234
5
Fig. 8.6 The relation between Nu and Re for model sheep without fleece---dashed line; the relation for the same model with flcece~fulliine and points (from McArthur and Monteith, IS/SOa).
Windspeed V(m 5-') Fig. 8.5 Relations between boundary layer resistance 'H and windspeed V for cylindrical bodies. Line a, smooth isothermal cylinder (McAdams, 1954); line b, cattle (Wiersma and Nelson, 1967); line c, sheep (Monteith, 1975); line el, sheep (McArthur
and Monteith, 1980a).
experiment, the heat loss from vertical subjects standing in a horizontal airstream was 8.5V t l. 5 W m-~ K- 1 (V in m S-I) and for a man with a characteristic dimension of 33 em, this is equivalent to A = O.7S, n = U.5 in equation (8.7). Although the corresponding values in Table A.5 are A = 0.24, n = 0.6, the two sets of coefficients give similar values of Nu for a relevant range of Re (104 to !O5), Similarity with values of Nu already quoted for horizontal cylinders illustrates the usefulness of non-dimensional groups for comparing heat loss from different systems. In a light wind, the heat loss from sheep and other animals may be governed by free convection, particularly when there is a large difference of
temperature between the coat and the surrounding air. When Merino sheep were exposed to strong sunshine in Australia, fleece tip temperatures rcached 85°C when the air temperature was 4SOC. With d = 30 em, the corresponding Grashof number is about 2 x lOR, so free and forced convection would he of comparahle importance when Re~ = 2 x Hf', i.e. when windspecd was about 0.7 m S-I. To calculate fleece: tempnatures in similar conditions, Priestley (1957) used a graphical method to allow for the transition from free to forced convection with increasing windspecd (see p. 124). For ere:ct humans, heat loss hy free convection is complex beGlliSC the Grashof number depends on the cube of the characteristic dimension. The nature of the convection regime therefore changes with height above the ground as shown in Fig. 8.7. The movement of air associated with free convection from the head and limbs has been demonstrated by Lewis and his colleagues using the technique of Schlieren photography (Fig. 8.8). The technique has also been used to study free convection from wheat ears and a rabbit (Fig. 8.9). The way in which the air ascending over the face of humans
134
Measurements of Convection
Heat Trallsfer
tt \
I ,
Turbulent flow
~
160
... .' .-':,' .. , .. .' - ..',,7:,~", -"':", . ,._ .' ..-.... , , . -
r·.:·. ., -
120
i iI
Transitional flow
E
o
Laminar flow
40
---,---
\
I
::
t
II
II
~
1 1
I
I
\
I
I I
I
:
i
II I
II
-
\
\
t
.
,
.
"
. ,~
,
-
"'" .• ,t. ,---
,
· .
,', -
.'
. ..
t +
I I
I
,
'
'
,.'
- - ---- -
t
11
i
-----~~----------------
';; '" 80-
.'.,',-
\ J.
I
~
..
.'"
'
•• ~:,,:--
--------------------------- - - -
--
r C' __"'~"< '..:., .. ',C' -.. '__ " - :- ' - ' . - . ' -.-""
.-c.'·; .-., ... _.;,.,. ..-. -.' . ... ~:"
'?, ." ,· .",,'','
-
, (
\ill ,
"
•
,
-:;,. .-
..
••
- - -",------'
-'
·
..
'.
'.
I
"'.
:-~'
• . ·,,• •• ·. - .. -
:'.' 7 .
;, :
,
"
· ::.
",
.-
, , · . /
.'
•
,,'
,
,
, ',:-. " . __ . ',., - • '., · ..'. , ;'"..... ':',' ... · '-." ,. . .:-' -5. {,'- "-.' ,,; , " -" ··.. .', -: ,-,' . ',-. ,' . , -- -' , ,', ....''' '. :-::'.: ;. ",,},-. . -, : .... -- '.-.--, ~:~ir:} ',','. ,,'::'-.' • , :-',-.--,:,':. l = NuLe m where m is 1/4 in the laminar regime and 1/3 in the turhulent r~gime. To calculate the Grashof number. it is convenient to replace the difference between the surface and air temperature To - T hy the difference of virtual temperature (p. 12). If e() and e are vapour pressures at the surface and in the
..
•
148
Mass Trallsfer
air and p is air pressure, the gradient of virtual temperature is
T,. - T,. "
I I
I
Reynolds Number
+ 0.38e,lp) - T(I + 0.38e1p)
~
To(i
~
(To - T) + 0.38(eoTo - eT)/p
100
500
1000
2000
1.6 r-"r----'="'-"------"=---'-~"----=~~-_;_;;__,
(9.5)
where temperatures are expressed in K. The importance of the vapour pressure term when T is close to To can be illustrat~d f 0.1, equation (10.2) should be used together with an estimate of Cd from Fig. 7.6 or equation (7.9) to find V, by trial and error. For example, to find the sedimentation velocity of a rain drop, radius 100 jJ.ffi, a first approximation can be estimated from equation (10.3), V, ~ 1.2 m S~l. The corresponding Reynolds number is 16 (confirming that equation (10.3) is strictly not applicable), and the drag coefficient (from equation (7.9» is about 3. The drag force on the drop (equation (10.1)) is
1
10
100
1000
1
10
Rep" 1000
"
(1!2)p, V/Cd1T? ~ 82 x 1O~9 N
0.5
which does not balance the gravitational force
o
(4/3)1Tr'pg ~ 42 x 10- 9 N
A smaller value of Vs must therefore be estimated and the drag force recalculated. The value of V, when drag exactly balances the gravitational force may then be found by interpolation, graphically or otherwise. This method is of general applicability. Figure 10.1 shows a specific case-the variation of Vs with radius for particles of unit density p = 1 g em -).
Biological aerosols such as pollen and spores can be treated as spheres of unit density and so the equations given here are adequate for finding their sedimentation velocities. Fig. 10.1 shows that Stokes Law (Rep"O.I) holds to about r ~ 30 flm (V, ~ 0.1 m S~l), a size range that includes most spores and pollen. Particles of soil and pollutant materials arc seldom spherical, and their density is seldom exactly 1 g cm- 3 (assumed in Fig. 10.1). Such particles are often characterized by their Stokes diameter, which is the diameter of a sphere having the same density and sedimentation velocity. Water drops with radius exceeding about 0.4 mm are appreciably flattened as they fall, and this increases Cd' Figure lO.l shows observed values of V~ for water drops. When r> 2 mm any further increase in drop weight is compensated for by an increase in deformation, and hence in Cd, and so Vs remain~ approximately constant. Raindrops much larger than 3 mm radius are seldom observed, as they tend to break up during their fall, but they may be maintained in strong updraughts in storm clouds.
1
-E ,~
-0.5 1
u
-1
->'
01
-a
E
0.1
'-.,-J -
1. 5
-a'" -2
001
0.01
-2.5
-3
0.001
-3.5
~40~-n';'----;--';";;----;\'--'~---;\'--O~--40.0001
NON·STEADY MOTION
0.5
If a particle obeying Stokes Law is projected horizontally in still air with velocity Vo at I ~ 0, its motion is subject to the drag force 61TvprV(t) where Vet) is the velocity at time t, and to the gravitational force mg. Resolvin¥ the motion into horizontal and vertical components, and writing dx/dt and d x/dr for the horizontal velocity and acceleration respectively, the equation of motion for horizontal displacement is
d 2x dx m dr ~ -61Tvpr dt
(10.4)
Similarly, for vertical motion, m
d'z
.2 ~
d.
dz
mg - 61Tvprdl
(10.5)
1
15
log
{. 2
}2.5
3
3.5
r (fL m)
Fig. 10.1 Dependence of sedimentation velocity on particle radiu:-. for :-.phcrical particles with density 1 g cm-;\, Stokes' Law applies for Rep tOO and Re = 10 respectively. Line C is fitted to experimental measurements with droplets. The plotted points are measurements with spores impacting on sticky cylinders as follows: Symbol r (ILm) R (mm)
+
x
V
2.3 1l.1-D.S
6.4
15 iO
0.1-D.8
6 15 3.3
..
15
0.4
"
15 1l.fJ9
The capture of mist (Fig. 1O.4(a), on spider's threads (R-O.I fLm) is even more efficient than the previous example because the drops (r = 10 fLm), much larger than the obstacle, are scarcely deflected, and any drops passing within a distance r of a thread are likely to be captured by interception. Interception is an important mechanism of deposition for particles with size comparable with or larger than the obstacles that they approach. Chamberlain (1975) and Chamberlain and Little (1981) reviewed the experimental data for particle deposition on vegetation. Stickiness, or wetness of surfaces, seems to be an important factor for the retention of impacting dry particles in the size range of spores and pollen (10-30 fLm radius) on leaves and stems. For example Chamberlain (1975) exposed barley straw to ragweed pollen (r about 10 fLm) at windspeeds of 1.55 m S-I and found that cp increased from 0.04 to 0.31 when the straw was made sticky. The presence or absence of a soft layer ·to absorb particle momentum on impact, thus avoiding 'bounce-off', probably explains these results. Bounceoff is most pronounced with large particles, high velocities (large momentum), small obstacles (thin boundary layer), and large coefficients of restitution at the surface (small loss of kinetic energy on impact).
Figure 10.5(a) shows the variation of F with KE; for Lycopodium and ragweed particles impacting on glass rods with diameters 3, 5 and 10 mm. For both types of particles, F was near unity for KE; < la-I' J then decreased by about 2 orders of magnitude as KE j increased to 10- 11 ]. The corresponding particle speed at the threshold for bounce-off was about 40 em S-I for Lycopodium and 70 em S-I for ragweed; the threshold kinetic energy for bounce-off was independent of particle diameter. Relative retention on wheat stems in the wind tunnel (Fig. 1O.5(b), filled symbols) did not decrease as steeply with KE; as for glass rods, but the threshold for bounce-off appeared similar to that for the rods. The greater scatter is probably a consequence of variability in surface structure of the stems. Bounce-off occurs when the kinetic energy of a particle after impact exceeds the potential energy of attraction at the surface. Surface structure is likely to influence both the energy absorption on impact and the potential energy of attraction, and so variation in bounce-off is likely with biologically variable surfaces. In the field (Fig. 1O.5(b), open symbols), the average value for Ffor a given KE; was smaller than in the tunnel. At low values of KE; this was probably because turbulent variation of windspeed about the mean caused KEi to range from sub-critical (constant F, no bounce-off) to well above super critical (F decreasing rapidly with KE,), and time spent at super critical KE; lowered the average value of F. At higher values of K E; this explanation is unlikely to apply, and the lower retention observed may then have been the result of strong gusts of wind re-entraining particles that had previously deposited. Table 10.1 Deposition of particles on segments of real leaves, and on filter paper, as a proportion of deposition on sticky artificial leaves made of PVC (from Chamberlain, 1975)
Relative deposition Diameter Particle
(ILm)
Grass
Plantain
Clover
Filter paper
Lycopodium spore Ragweed pollen Polystyrene Tricresylphosphate Aitken nuclei
32 19 5 1 O.OS
0.45 0.15 1.74 1. 70 1.1l6
0.26 0.11 1.82 2.60 1. 70
0.18 0.23 3.25 5.51l 0.86
0.70 0.68 1.98 6.40 1.54
Sticky pvc 1.00 1.00 1.00 I.(X)
1.00
.. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. lilt ..
172
Steady Motion
Mass Trallsfer
173
( a) I
I
c
--c0
I
H-
-
OH-
-
~ ~
o
~
~
>
o
0.01 f-
00
0
0
o o
0
-
0 0
~
a:
o o 'b
OOOH-
o
0
o
0
O. 0001 "---:-:-_~':--_L.--,I:--_1---, ',----_1---, '-c-10-14 10- 13 10- 12 10-11 10-10 10-9 Kinetic energy (J) ( b) I
c
-0
0.1
_
c
00
. '. "II,.,,,, .ih· . ';,
.,.
1f- ••
• • •
-
~
0
0
""
-
cP-
@/I,~
dO
0
~
>
o.
00 0 0"
0.01 I-
-
• • • ~
0
~ ~
~
to ••
0
~
I
0
•
-•
0
0 0
-
0
~
a: 0.001f-
I
,
,
0.0001 L--;-;,3;---"---;,;;-2--L;;;-----L;-;;-----.l 101010- 11 10-10 10- 9
Fig. 10.4 (a) Drops of fog which have impacted on and been intercepted by threads of a spider's web. The threads of the web are at least an order of magnitude smaller than individual drops (typically 10 fJ.ffi diameter) and so are efficient collectors by interception (p. 170). (b) Drops of fog collected on an author's beard after cycling. Human hair is about 50 j.lffi diameter, so the drops visible in this picture must have coalesced from many impacted fog drops.
Kinetic energy (J)
Fig. 10.5 Variation with kinetic energy at impact of the relative retention for ragweed (0) and Lycopodium (0) particles impacting (a) on glass rods, diameters 3,5 and 10 mm, in a wind tunnel, and (b) on wheat stems 3-4 mm diameter in the wind tunnel (filled symbols) or in the field (open symbols).
174
Mass Transfer
Steady Motion
In contrast to the factors influencing capture of particles larger than about
..
•
10 fLm, the capture of small particles (r
100
.. .. .. .. .. II
• • -
-,., U>
E E
10
--.,
.u-
,, ,
0
> c
-
.-0 .U>0
.,
c.
• B',
~
•,
,, ,
1.0
A
•
,,
0
.,
~•,
,
• ./
,,
0.01
0.1
1.0
10
Particle radius (fLm)
II III
sedimentation velocity, demonstrating that sedimentation accounted for a large proportion of the deposition only when particles exceeded IO j.l.m radius. Considered together, deposition processes are least efficient for particles of about 0.1-0.2 !-Lm radius, and it is significant that man-made aerosols in this size range are found widely distributed in the Earth's atmosphere. In particular, sulphate particles in this size range, formed by the oxidation of sulphur dioxide, can he carried extremely long distances in the atmosphere, and tend to persist until they encounter conditions of high humidity in which they can grow by delinquescence and condensation into larger droplets which are more effectively deposited. The figures in Table 10.2 are also relevant to the hazards of dust inhalation. During nasal breathing in man, particles larger than 10 J-Lm are efficiently Table 10.2 Characteristic distances for particle transport Particle Radius r (!-lm) Stopping distance (!-lm) given initial velocity of 1 m S-l Distance (!-lm) travelled in 1 second by virtue of: Terminal velocity Brownian diffusion
0.01 14x 10-.1
o. I
1
JIJ
0.23
13
-.-
128
14 x 10- 2 00 160 21
5
1230
1200 1.5
impacted and intercepted from fast moving air in the nose; as bronchial tubes divide, the average air velocity in each tube decreases, but particles from 1-10 J.1m have a strong probability of impaction when the air stream changes direction rapidly on branching, and they also sediment onto the walls of non-vertical tubes. Particles smaller than 1 /-Lm are most likely to be deposited deep in the lungs in narrow bronchioles and alveolar sacs where the distance to a site of deposition is comparable with the distance which they can travel by Brownian diffusion during the time that they arc within the lungs.
•
I
175
Fig. 10.6 The variation with particle radius of the deposition velocity to short grass exposed at a windspeed of about 2.5 m S-l (from Little and Whiffen, 1977). Line A is the sedimentation velocity of particles with density 1 g cm-\ line B was calculated from Brownian diffusion theory.
STEADY STATE HEAT BALANCE (i) Water surfaces and vegetation
The heat budgets of plants and animals will now be examined in the light of the principles and processes considered in previous chapters. The First Law of Thermodynamics states that when a balance sheet is drawn up for the flow of heat in any physical or biological system, income and expenditure must be exactly equal. In micrometeorology, radiation and metabolism are the main sources of income; radiation, convection and evaporation arc methods of expenditure. For any component of a system, physical or biological, a balance between the income and expenditure of heat is achieved by adjustments of temperature. If, for example, the income of radiant heal received by a leaf began to decrease because the sun was obscured by cloud, leaf temperature would fall, reducing expenditure on convection and evaporation. If the leaf had no mass and therefore no heat capacity, the drop in expenditure would exactly balance the drop in income, second by second. For a real leaf with a finite heat capacity, the drop in temperature would lag behi:1d the drop in radiation and so would the drop in expenditure, but the First Law of Thermodynamics would still be satisfied because the income from mdiant heat would be supplemented hy the heat given up by the leaf as it cooled. This chupter is concerned with the heat balance of relatively simple systems in which (a) temperature is constant so that changes in heat storage are zero; and (b) metabolic heat is a negligible term in the heat budget. The heat budget of warm-blooded animals, controlled by metabolism, is considered in the next chapter and examples of diurnal changes of heat storage are considered in Chapters 13 and 15.
HEAT BALANCE EQUATION The heat balance of any organism can be expressed by an equation with the form Rn+M = C+AE+G (III) where the bars indicate that each term is an average heat flux per unit surface area. (In the rest of this chapter, they are implied but not printed.) In this context, it is convenient to define surface area as the area from which heat is lost by convection although this is not necessarily identical to the area from
",-
178 Steady State Heat Balance
Heat Balance Equation
179
which heat is gained or lost by radiation. The individual terms are
Rn =
net gain of heat from radiation M = net gain of heat from metabolism C = loss of sensible heat by convection AE = loss of latent heat by evaporation G = loss of heat by conduction to environment The conduction term G is included for completeness but is negligible for plants and has rarely been measured for animals. . . . The grouping of terms in the heat balance equatIOn IS dictated by the arbitrary sign convention that fluxes directed away from a surface are positive. (When temperature decreases with dista~ce z frot.TI. a surface. so that aT/az-
28
\
I
Ii -
I
'.I
1.0
::0 ~
-
24
co ~
-
E C hotter than the surrounding air. Greater excess temperatures are observed on very large leaves in a light wind because rH is large. From the same set of calculations, it can be shown that the relative humidity of air in contact with the epidermis will usually be similar to the relative humidity of the ambient air, and this feature of leaf microclimate may have important implications for the activity of fungi which need a very high relative humidity to reproduce and grow.
Dew When Rn is negative at night, condens if it is impermeable to radiation of all wavelengths, and if the evaporation of sweat is confined to the surface of the skin with mean temperature T" the flux of sensible heat through the coat is
M - (AE, + AEJ = pcr(T, - To)!r, I
(12.4)
The flux of heat through the skin surface is ( 12.5)
I
where TJ is the resistance of the body tissue and Tb is the body temperature. Eliminating To and T, from equations (12.3) to (12.5) gives the relation between heat balance components as
M + (rHR/r,)R n; = PCp(Th - T)/r, + AE, + AE,(rIiR + r,.)/r,
(12.6)
where T, = THR + 'c+ 'h' This equation can be used to explor,· the physiologicany significant air temperatures shown in Fig. 12.4 if it is wntten in the form T
=
To - {r,M + rHRRn; - r,AE, - (rliR + r,)AEJ/pc r
(12.7)
Equations (12.6) and (12.7) will now be used to' examine five discrete regimes incorporated in the thermo-neutral diagram.
At an air temperature of T 1, sometimes referred to as the 'cold limit', the body produces heat at a maximum rate Mmux. Assuming that the sum of the two latent heat terms in equation (12.7) is O.2Mm" (see p. 201), the value of T J is given by (12.8)
Table 12.1 contains values of measurements of TJ summarized by Mount (1979) for newborn and mature individuals of three species. Comparisons emphasize the value of hair in establishing the size of " and therefore of the product r,M max ' Estimates of this quantity given in the Table were derived by assuming that Tb = 37°C for all species, that Rni was zero in the experimental
conditions and that pC r = 1200 J m -.1 K-'. When air temperature falls below the cold limit, heat is dissipated faster than it is made available by metabolism and radiation so that the steady-state heat balance represented by equation (12.7) cannot be achieved and body temperature must then decrease with time (see p. 177). This depresses the metabolic rate so that temperature falls still further, leading to death by hypothermia if the process is not reversed. II
T, < T< 1'2
In this regime, sensible heat loss decreases linearly with increasing air temperature, inducing an identical decrease in metabolic rate provided the latent heat component of the heat balance remains constant. Differentiation of equation (12.6) then gives iJM aT
(12.9)
If r, = 600 s m-', for example, aMlaT- 1 is -2 W m-' K-'. Equation (12.9) is valid between the cold limit TJ and a temperature T, at which the metabolic rate reaches a minimum value M min , usually somewhat larger than the basal metabolic rate (p. 200) on account of physical exertion and/or the digestion of food. Repeating the assumptions used to find an approximation for TI gives (12.10)
Observed values of T2 and estimates of ',M rnin are in Table 12.1. Livestock exposed to a temperature below T2 need more feed to achieve the same livcweight gain or production of milk, eggs, etc. than a comparable animal kept in the thermo-neutral zone. The temperature T2 is therefore referred to as the (lower) critical temperature.
Specification of the Environment
208 Steady State Heat Balance
209
III TzT,
V
(M-AE)=pcr(T,,-T,)/r'lR
As temperature rises further, loss of latent heat cannot increase indefinitely. The rate of evaporation of sweat is limited by the aerodynamic resistance of the body and therefore by winds peed and the rate of evaporation from the respiratory system is likewise limited by the rate of panting and the volume of air inhaled and exhaled. Both rates of evaporation depend on the vapour pressure of the atmosphere. When AE reaches a maximum (or even at a somewhat lower temperature as Fig. 12.4 suggests), body temr"ature starts to rise, a primary symptom of hyperthermia. Differentiating equation (12.7) with respect to T when AE is constant gives 1=
T-"h aT
.:.:iI
aM aTb r,
,
ilTh ilT pC r
or
aT -----"h = {1-r,(aMlaTh)/pcrl-' aT The requirement that aTt/aTshould be positive implies that ilM ilTb
-0.1'" Ri" I
(14.14)
and F ~ (1- 16Ri)o75
Ri Z2, etc.) if the resistances across these potentialS are also known. Within the canopy, resistances corresponding to the stomata and boundary layers of individual leaves have a clear physical significance, but the validity of describing transfer in the air within the canopy by Ohm's Law analogues is discussed later in this chapter. It is possible to derive a parameter which plays the same part in equations for the water vapour exchange of a canopy as the stomatal resistance plays in similar equations for a single leaf. This parameter will be given the symbol re. where the subscript denotes canopy, crop or cover. It was shown in Chapter 14 that the sensible heat loss from a surface can he written in the form C = -pc"lI,'(aTlall)
where T is a linear function of u. As a special case, the gradient iJT/iJu can be written [T(z) - T(O)]/[u(z) - 0] where T(O), obtained by the extrapolation shown in Fig. 15.2, is the air temperature at the height where u = 0, i.e. Z = d + Zo- The above equation can therefore be written as C = -pc p ll,2[T(z) - T(O)]/u(z) (15.1)
I
246
Resistance Analogues 247
Crop Mierometeorology
I
----
I
I
--
I
I
Resistance
" 'z
"
{ ln
lz-dl}/u.k
+ ,I
"
I
Stomata
Boundary layer
•
'5
•
_____ - - - - 7 ;
'a
/ / /
/ /
,.-
t
I
,
t
/
/
/
~.-----zz
/
/ /
Iln z 61lu.k
/
L
"~----"
, T
t101'
TlOI WlOdspeed or temperature
Fig. 15.1
where
Fig. 15.2 Diagram :..howing how T(O) and T(O') are determined by plotting temperature against [lo(z - d)j/u*k. 1'(0) i~ the value of temperature extrapolated to z - d = Zo and nO') is the value extrapolated to z - d = zo'. The significance of r" and rh is shown on the left -hand axis and is discussed on p. 248.
Resistance model for a plant in a stand of vegetation.
u(z)lu* 2 can be regarded as an aerodynamic resistance between a fictitious surface at the height d + zo, and the height z. Similarly, it c~n be shown that 'aH =
AE~
_ ,--PC",r {e(z) - e(O)}
(15.2)
where e(O) is the value of the vapour pressure extrapolated to u ~ 0 and raY = YaH = u(z)/u* 2 . The diffusion of water vapour between the intercellular
spaces of leaves and the atmosphere at height z can now be described formally by the equation
AE ~ -per {e(z)-e,(T(O»} 'Y
raV+rc
This relation defines the canopy resistance rc and is identical to the corresponding equation for an amphistomatous leaf with Tc replacing the stomatal resistance r, (p_ 188) and r,v replacing rv. Introducing the energy balance equation, writing roll ~ r"v ~ r" and eliminating T(O) (see pp. 185-187) gives AE ~ .:I.(R" - G) + perk( T(z») - e(z))/r" .:I.+-y'
( 15.4)
where "'{* = 'V(ra + rJlra - Values of 'I,.; for a given stand can therefore be derived either directly from profiles of temperature, humidity and windspeed using equations (15.1) and (15.2) or indirectly from the Penman-Monteith equation (15.4) when the relevant climatological parameters are known and
AE is measured or estimated independently. Two objections can be raised to this apparently straightforward method of
(15.3)
separating the aerodynamic and physiological resistances of a crop canopy. In the first place, the values of 'c derived from measurements are not unique
248
Resistance Analogues 249
Crop Micrometeorology
unless the sources (or sinks) of sensible and latent heat have the same spatial distribution. In a closed canopy, fluxes of both heat and water vapour are dictated by the absorption of radiation by the foliage and. provided the stomatal resistance of leaves does not change violently with depth in the part of the canopy where most of the radiation is absorbed. the distributions of heat and vapour sources will usually be similar but will seldom be identical. Conversely, anomalous values of 'c are likely to be obtained in a crop with little foliage if evaporation from bare soil beneath the leaves makes a 50ubstantial contribution to the total flux of water vapour. In the second place. the analysis cannot yield values of rc which are strictly independent of unless the apparent sources of heat and water vapour, as determined from the relevant profiles, are at the same level d + Zo as the apparent sink for momentum. This is a more seriolls restriction. Form drag, rather than skin friction (p. 103), is often the dominant mechanism for the absorption of momentum by vegetation so that the resistance to the exchange of momentum between a leaf and the surrounding air is smaller than the corresponding resistances to the exchange of heat and vapour which depend on molecular diffusion alone. It follows that the apparent sources of heat and water vapour will, in general, be found at a lower level in the canopy than the apparent sink of momentum, say at z = d + zo' rather than at z = d + Zo where zo' is smaller than zoo Atmospheric resistances to transfer may therefore be described in terms of 'a, the resistance to momentum transfer and 'h, an additional resistance, assumed to be the same for heat and water vapour. From the definition of ra and the wind profile equation in neutral stability
0.3
4
E --, ~
'~
'a
r,
0
Crop
Short grass
'a
=
In[(z-d)!z,,']
ku*
3
1
- 20 - 30 50 100 5
Wlndspeed (m s-l)
Fig. 15.3 Calculated values (from equation (15.5» ?f resistance fa in relation to windspeed over surfaces with roughnesses charactenstlc of: short grass (zo = 1 mm, d ~ 7 mm); cereal crop (za ~ 0.2 m, d ~ 0.95 m); forest (20 ~ 0.9 m, d ~ 11.8 m). Wind~peeds are referred to a standard height z - d = 5 m for each surface.
(15.5)
In near neutral and stable conditions the resistance derived from equation (14.23) is
In the derivation of this expression the effective height for the sink of momentum is z = d + z" for which (z - d)/zo ~ 1 and so r, = n. Similarly, the resistance between a height z above the ground and the apparent source (or sink) of heat and vapour at a height d + z,,' can be written as ,
-10
o1
{In[(z-d)!zolJ!ku.
~ {In[(z - d)/zoW!k 2 u(z)
,
,
~ pu(z)!.. ~ u(z)!u. 2 ~
5
02
=
In[(z- d)/zo]
ku*
In[(z,)zo']
+ --'''--''---''-' ku*
z-d
1
[ku.r
In
Zo
+
5(z-d)
(15.7)
L
provided that z - d is at least an order of magnitude greater than zoo When [In(z - d)/zo]!ku, is identified as ra> the additional component +[5(z- d)IL]/ku. can be regarded as a stability resistance. Similarly, equation (14.11) can be used to show that the stability resistance in more unstable conditions (L negative) is approximately [4(z-d)IL]!ku •. The tendency for 1/r. to become independent of windspeed as the speed decreases is more pr~lnounced in unstable than in n~utral
